Addition frames are used to help children with the intermediate step of translating concrete models into numbers by recording the tens and ones that the models represent.

The ten frame shows the place of each digit in the numbers being added. Children must be sure to write the digits in the appropriate place in the chart. They must be especially careful when they add a 2-digit number to a 1-digit number, being sure to place the 1-digit number in the ones column. It is useful to discuss why this placement is important and why putting the single digit into the tens column changes the sum and makes it incorrect.

Another potential error is when children add a number with no ones, such as 30, and do not place a 0 in the ones place. Children must remember to include the zero as a placeholder and not to write the 3 in the ones place. Again, a discussion about the reasons for writing the number 3 in the tens place will help solidify place-value concepts and reinforce the application of the 2-digit addition algorithm.

After children have written the problem correctly in the addition frame, they can more easily see how to add down the columns, to find the correct sum, starting on the right with the ones. It may be difficult for children to remember to add the ones column first, since they have learned to read from left to right rather than right to left. Provide ample practice, reminders, and reinforcement to solidify this pattern. Children will also need practice when they begin to regroup ones as tens. Drawing or making a model to the right of the addition frame will help.¹ Children can also model a problem to the right of the digits to check an answer. Remind children that checking an answer is always a good idea. You might wish to informally introduce the Commutative Property of Addition as a way to check if children are ready to add without models.

The benefit of working with manipulatives to reinforce place-value concepts and build representations of regrouping routines. Children will benefit from practice with the following activity.

Separate children into pairs and have them practice modeling subtraction problems. Write 42 − 18 on the board. Have children build 42 (4 rods and 2 cubes) and challenge them to generate strategies for "taking away" 18. Children should see that they can "trade" 1 rod for 10 cubes and then take away 18 (1 rod and 8 cubes). The remaining rods and cubes show the difference.

You can also use the base ten blocks to reinforce children's understanding of the relationship between addition and subtraction. Show children how to use the blocks to generate the appropriate "add to check" addition problem—in this case, 24 + 18 = 42. Invite a volunteer to demonstrate how the total number of cubes (4 + 8 = 12) should be traded for 1 ten rod and 2 cubes.

Challenge children to create with their own pairs of related addition and subtraction problems to model and share.

Children extend the use of ten rods and ones units for modeling 2-digit numbers to include hundreds flats for modeling 3-digit numbers.

Give pairs of children base ten blocks to work with and review the value of hundreds flats, tens rods, and ones units.

Because many base-ten blocks do not come apart, some children find it hard to understand that ten ones are equivalent to one ten, or ten tens are equivalent to one hundred. Even children who have learned when to trade have not necessarily made the connection that ten "tens" equal one "hundred". Take time to help children make the connection by having them line up ten "ones" side by side with 1 "ten". Have them count the separate units and count the fused units on the ten rod. Have them make the connection that the single ten rod is equal in value to the ten individual ones units. Do the same with tens and a hundreds flat.

Ask volunteers to name a 3-digit number and challenge one partner to model the number and the other partner to check the model. Children should alternate roles as other numbers are presented to the class.

Coin manipulatives enable children to examine each coin's attributes closely and to estimate a quantity of money based on a visual scan of a handful of coins. Making amounts in different ways develops flexible thinking, practices the value of coins, reinforces place-value concepts, and is a practical skill that motivates children to learn coins' values.

Children will benefit from using coins in an activity such as the following: First demonstrate to children how to separate a group of coins in three ways to represent the same value. For example, show them how to show 27 cents in three different ways as illustrated below.

Then, give pairs of children a selection of coins in order to play "Coins Three Ways." After giving children a coin amount, challenge them to "build" that amount of money in three different ways as you did in the demonstration.

As an additional challenge, pose a question such as: "I have 35 cents in my pocket made up of 4 coins. What are they?" Encourage children to work with their manipulatives to figure out what the mystery coins are (in this case, 3 dimes and 1 nickel). You can separate the class into groups of two to play this game with each other. Encourage children to document their work by drawing circles to represent the coins and labeling the center of each coin with a Q (for quarter), N (for nickel), D (for dime), or P (for penny).

Although the main place-value models used will be base-ten blocks, children can use connecting cubes to understand the meaning of a hundreds flat. Working with cubes to build a hundred helps children see that a hundreds flat represents ten groups of ten and provides a good transition to pregrouped base-ten blocks.

Have children work in small groups to connect ten cubes to make rods of ten. Alternatively, if you feel that children are secure in their understanding that a ten rod represents ten and do not need the assurance of putting together ten individual cubes, allow them to use ten rods in place of the cubes. For this activity, each group will need cardboard (or thick paper), scissors, two rubber bands, and at least 100 connecting cubes or ten tens rods.

Have children lay out ten rods on the cardboard, cut the cardboard to the size of the rods, and then use two rubber bands to secure the rods to the cardboard.

Have each group label the back of the cardboard 100. Then, elicit other names for 100 (10 tens, 100 ones, hundred.) Have children add these labels.

Pass around a base-ten hundred flat to each group and have children compare and contrast the two models. Some determined children may even count the "squares" in the hundreds flat to confirm that it has the same number of units. Have them share their results with the class.

Connecting cubes help children visualize tens and ones in a two-digit number because they allow singles to be grouped as tens. The grouping most clearly reflects the relationship between ones and tens. The following activity will help children use their connecting cubes to model a two digit number.

Help children model the number by having them identify the digit in the tens place and connect that many cubes into groups of ten. Then, have them identify the digit in the ones place and lay out that many individual cubes. Have children practice by writing the number 43 and then modeling this number. Check the models for accuracy and repeat with other numbers.

When children have demonstrated fluency with the skill, have them work in pairs. One child tosses a number cube two times to make a 2-digit number. The other child uses connecting cubes to model the number. Children work together to check the model. They alternate roles and repeat the activity.

To identify a number represented by a model, children can count the total number of cubes. This brute force method is time-consuming, and has a high error rate. Instead, the weaker "counting" strategy should be substituted for the more fluent place-value concept. Children should group the cubes into tens and ones and then count the groups to find the number. At some point in the activity, you may wish to relate a group of ten connected cubes to a ten rod and note the similarity. Children may count the demarcation in the ten rod to assure themselves that it represents ten in a more efficient way than connecting ten cubes.

Finally, have pairs work with the rods and unit cubes to represent numbers and write the number on a place-value chart.

Dominoes are useful for modeling related facts because they can represent the addition and subtraction facts through 6 + 6. The center divider keeps the parts distinct, yet the dots are clearly part of a single whole. In subtraction situations, children sometimes have a tendency to focus only on the number left or the answer and return the number taken away to a supply pile. This maneuver prevents children from reflecting on the situation and reversing it. With dominoes, since both the whole and parts are shown, children can use the following activity to experience the relationship between addition and subtraction.

Distribute dominoes and have children note and record an addition fact associated with their domino, such as 5 + 4 = 9. Then have children turn the domino in the opposite direction and note and record the related addition fact, 4 + 5 = 9

Next, have children cover one side of the domino with a finger or index card and write the related subtraction fact. Repeat for the other subtraction fact by covering the other side of the domino. Ask: How much is left when you take away 5? How much is left when oyu take away 4? Have children say and record the related subtraction facts:
9 – 5 = 4, 9 – 4 = 5.

After organizing the four written number sentences, have children use highlighters of different colors to mark each number (all of the 5's, for example, could be yellow). Help them see how the placement of the number changes and connect the relationship between the whole and the parts in the addition and subtraction sentences.

Relate the different written forms of notation (vertical and horizontal formats) by having children rewrite their number sentences in the other format.

To develop a deeper understanding of how and why one measures, children should be given many opportunities to measure and compare real objects with both nonstandard and standard units. Children at this level of development should become familiar with the use of rulers, scales, liquid cup measures, and thermometers, as well as their related units of measure. The following activity is designed to help them develop this familiarity.

Give each child an inch ruler and a pile of paper clips. The class will also need some classroom scales for this activity.

Ask each child to choose a classroom object and measure its weight and length (and width or height, if appropriate) in both standard units (inches) and nonstandard units (paper clips). Have each child record the data about the object in a chart.

Have children take turns telling the object data to their classmates as a "Who Am I?" riddle. For example, a student might say, "My object is a rectangle 12 inches long and 9 inches wide. It is 9 paper clips long and about 6 paper clips wide. It weighs 3 pounds. What is it?" Encourage other children to use their estimation and logical reasoning skills to deduce a range of possible answers.

Children can collect and compare all of the data and then order the objects by length, weight, or other attribute to give a sense of how objects compare to each other.

Using one-centimeter graph paper to create arrays is helpful for children both in understanding the relationships of multiplication to adition and multiplication to division and in understanding the relationship of those concepts to the algorithm.

You can begin by copying 1-centimeter graph paper onto an overhead transparency and coloring in a 10 × 10 array. Cut a large L out of construction paper. Use the L to change the array displayed and have children name the array shown as "3 times 4".

Next, divide the class into small groups of four or five children and give each group enough butcher paper to make an 11 × 11 grid. Guide children in making the grid out of meter sticks. Then, have them label across the top and left side so that each column and row is numbered 1 to 10 in order.

Give each group scissors, glue, 1-centimeter graph paper, and crayons or markers. Have the group work together to make an array for each multiplication fact, cut out the array, and glue it on the correct spot on the chart. This project can take up to five lessons for children to complete, and is extremely helpful in giving children an array to visualize for each multiplication fact to 10 × 10. At some point in the activity, help children to recognize that the same array that shows a fact such as 3 × 7 = 21 can also show that 7 × 3 = 21, giving them early contact with the Commutative Property of Multiplication.

Pattern blocks help children explore patterns in more depth. Children can use the blocks to reinforce shape names, identify figure properties, and build and extend patterns. Children in second grade should be able to explore repetitive patterns in both two and three dimensions. Children need to be able to name the attributes that make up a pattern as well as to identify when a pattern has been broken.

Translating a shape pattern into an alphabetical or numerical pattern can help children begin to recognize abstract patterns as well as concrete patterns. To help with this understanding, provide children with a selection of pattern blocks. Then write a letter-based pattern on the board, such as "AABAAB." Challenge children to use their blocks to build a similar pattern. For example, a child could build a pattern with a square, a square, and a triangle, followed by a square, a square, and a triangle.

Write more letter patterns on the board, and challenge children to show these patterns with pattern blocks.

Finally, extend the concept by reversing it. Have children build a pattern out of pattern blocks. Then, challenge them to represent the same pattern using an alphabetical format or a numberical format.

Children in Grade 2 should experience probability in an informal, pictorial, and practical way. Probability concepts should relate to the likelihood of real-world events, using terms such as more likely and less likely. Spinners are particularly useful for helping children visualize probability concepts. The following activity will help children solidify their understanding of probability concepts.

Separate the class into teams of two or three. Give each team a spinner divided into three unequal parts. Ask children to describe their spinners and compare the amount of space devoted to each color.

Have each group predict which color will “win” when they spin the spinner 20 times. Direct teams to spin the spinners and record the data on a chart such as:

After children have recorded their data, invite them to discuss the results. Prompt them by asking which color on their spinners "won" and by having them explain why they think that color "won". Some children will respond that the color "won" because the spinner landed on it the most times. Guide these children to understand that the more space a color takes up on the spinner, the greater the likelihood that the arrow will land on it, and the greater the likelihood that it will "win."

Have children repeat the activity with different spinners. When children use a spinner with equal-size sections, help them understand that the probability of landing on a particular color is the same for any other color (though the results may not be exact for each color).

Then, challenge each group to divide and color a spinner so that blue is most likely, red is least likely, and yellow and green are equally likely. Have them test their results by spinning the spinner 30 times.

Second graders have had experience sorting objects in a set by attribute. Sorting objects by attribute in a venn diagram helps children sort by two or more attributes and gain a visual sense of the groups and how the attributes intersect. You can attribute blocks and circles on the overhead projector to introduce children to sorting objects in a venn diagram. Focus the task by providing a limited number of attrivutes to sort—providing too many choices may result in a venn diagram with no intersection.

Real graphs, such as concrete graphs, use the actual objects being sorted and graphed. Most of the graphs that children work with in Chapter 3 are symbolic graphs, which use, for example, squares, blocks, tallies, or X's to represent the things being counted. Sorting actual objects into venn diagrams made of yarn is an excellent preparation for building a real graph and prepares children for the symbolic activities throughout the chapter.

Have children sort a variety of red shapes and different color triangles into two overlapping yarn circles. They can work with partners or small groups. Allow time for children to discuss different ways to sort the shapes. Challenge them to find a shape that belongs in the overlap area and discuss why this shape belongs there.

Then, have children sort a collection of coins and challenge them to come up with their own attributes. Help chilren label their circles and have them share their results with the class as a method of checking student work.

Finally, give children a set of numbers and challenge them to notice the attributes of the numbers in the set. Children may sort by odd and even numbers or by skip-counting numbers (e.g., counting by 3's). Have children label their circles and share their results with the class.