Use pairs and equal rows to explain totals

Learning goal: Identify odd and even groups up to 20, use addition to count rectangular arrays up to five rows and five columns, and partition a rectangle into rows and columns of equal-size squares. The models are related, but pairing objects, counting an array and partitioning area are distinct tasks that each require practice.

An even group can be separated completely into pairs. Twelve counters make six pairs with none left over. The same twelve can be split into two equal groups of six, giving 12 = 6 + 6. An odd group leaves one unpaired when all possible pairs are made. Thirteen counters make six pairs and one extra.

The drawing places twelve dots in three rows of four. Each row has the same number of objects and each column is aligned. This arrangement is a rectangular array. Count by rows: 4 + 4 + 4 = 12. Count by columns: 3 + 3 + 3 + 3 = 12. The two equations count the same objects in different groupings; their totals must agree.

Rows usually run across the page and columns up and down in this displayed model. Turning the whole array changes its orientation, not its total. Always identify how you are grouping before stating an equation. “Three rows” alone does not establish the total unless the number in each row is also known.

Do not count spaces between objects as if they were additional objects. An array of three rows and four columns contains twelve dots, not the twenty points where grid lines cross if someone draws a three-by-four grid of boxes around them. Explain the unit being counted: dots for the object model, or square regions for the partitioned rectangle.

To partition a rectangle, draw two internal horizontal cuts to make three equal rows and three internal vertical cuts to make four equal columns. If the cells are all same-size squares, there are twelve square regions. The outside boundary encloses the whole rectangle. Counting the cuts gives two and three, but those are not the numbers of resulting rows and columns.

Check that the drawn cells are squares, not merely equal-looking rectangles. A three-by-four array of equal squares makes an outer rectangle whose width is four square side lengths and height is three. Drawing a square outer boundary and splitting it into three rows and four columns would create nonsquare cells instead.

Use equal addends to explain evenness too. Eighteen can be split into nine and nine, so 18 = 9 + 9. Seventeen cannot be completely split into two equal whole-number groups without one left over. Pair physical objects or a reliable model before relying only on the final digit pattern.

Worked example 1: Pair a group. Arrange thirteen counters into pairs. Six pairs account for twelve counters and one remains. Thirteen is odd. Remove the extra counter and twelve is even, expressible as 6 + 6.

Worked example 2: Count the array twice. Three rows of four give 4 + 4 + 4 = 12. Four columns of three give 3 + 3 + 3 + 3 = 12. Both account for every dot exactly once.

Worked example 3: Count cells rather than boundaries. A rectangle split into three equal rows and four equal columns of squares has twelve cells. Two internal horizontal lines create three rows; the number of lines and the number of regions differ.

Guided practice: Pair groups from zero through twenty over several short sessions, explaining leftovers and writing equal-addend equations for even groups. Build different arrays no larger than five by five and write a repeated-addition equation. Then draw or assemble an equal-square partition of a rectangle.

Independent tasks: Determine the parity of a supplied group, write an even number as two equal addends, count an unfamiliar array and construct a rectangle partition with a specified number of equal-square rows and columns. An already drawn grid does not demonstrate that the learner can construct one.

Feedback: If a learner counts internal lines, touch each region instead. If array rows have unequal numbers, rearrange and discuss why the original was not the intended rectangular array. Credit explanations that keep the counted units clear.