Big Idea(s):

Grouping items in 2s, 5s, and 10s

Expectations:

B2. Operations: use knowledge of numbers and operations to solve mathematical problems encountered in everyday life

• Multiplication and Division: B2.6 represent division of up to 12 items as the equal sharing of a quantity, and solve related problems, using various tools and drawings

C2. Equations and Inequalities: demonstrate an understanding of variables, expressions, equalities, and inequalities, and apply this understanding in various contexts

• Variables: C2.1 identify when symbols are being used as variables, and describe how they are being used

teacher background

Students may benefit from prior experience with:

• skip-counting forward by 2s, 5s, and 10s

• counting collections by grouping

• decomposing numbers and objects into equal groups, with and without leftovers

• demonstrating the principles of counting:

  • one-to-one correspondence – says one word for each object or group of objects counted;

  • conservation of number – rearranging the objects in a set does not change the quantity;

  • cardinality – the last counting word tells “how many” objects are in the set;

  • stable order – the counting numbers are always said in the same repeatable order

Key concepts

• Division, like multiplication, can describe situations involving repeated groups of equal size.

• While multiplication names the unknown total when the number of groups and the size of the groupsare known, division names either the unknown number of groups or the unknown size of the groups when the total is known.

Note

• The inverse relationship between multiplication and division means that any situation involving repeated equal groups can be represented with either multiplication or division. While this idea will be formalized in Grade 3 (see B2.1), it is helpful to notice this relationship in Grade 2 as well.

• While it may be important for students to develop an understanding of these operations separately at first, it is also important for students to observe both multiplication and division situations together, to recognize similarities and differences.

• There are two different types of division problems.

Equal-sharing division (also called “partitive division”):

• What is known: the total and number of groups.

• What is unknown: the size of the groups.

• The action: a total is shared equally among a given number of groups. Equal-sharing division is also being used to develop an understanding of fractions in B1.6 and B1.7.

Equal-grouping division (also called “measurement division” or “quotative division”):

• What is known: the total and the size of groups.

• What is unknown: the number of groups.

• The action: from a total, equal groups of a given size are measured out. (Students often use repeated addition or subtraction to represent this action.)

Equal-group situations can be represented with objects, number lines, or drawings, and often the model alone can be used to solve the problem. It is important to model the corresponding equation(addition or subtraction and division) for different situations and to make connections between the actions in a situation, the strategy used to solve it, and the operations themselves.

Show or project 8 linking cubes.

Ask, “How many ways can we arrange the cubes in equal groups with no leftovers?”

Have volunteers model different ways.

Ask: “How many groups of 2 did we make? How many groups of 4? Can we make groups of 3? Why or why not?”

Give each group three bags of items and a recording sheet (Master 100). Have ten-frames (Multi-Use Card 1) available.

• Each of you take one bag. Count the items in the bag.

• Can you group the items in 2s with no leftovers? If you can, write the number in the chart.

• Can you group the items in 5s with no leftovers? If you can, write the number in the chart.

• Can you group the items in 10s with no leftovers? If you can, write the number in the chart.

• Count small collections of items in the classroom. Can you group the items in 2s, 5s, and/or 10s with no leftovers? If you can, write the number.

LOOK-FORS:

• How do students count the items (e.g., by 1s, using skip-counting)?

• How do students organize their items (e.g., using ten-frames, in piles, in ordered arrangements)?

• Do students recognize that the quantity will be the same no matter how the items are grouped?

• Do students recognize that not all quantities can be grouped in 2s, 5s, and/or 10s?

How to Differentiate:

Accommodations: Give students bags of 5, 8, and 10 items.

Extension: Have students decide if a number can be grouped in 2s, 5s, and/or 10s without using concrete materials.

Combined Grades Extension: Use bags of 20, 25, and 32 items. Have students put the items into equal groups in as many ways as they can.

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