Standards-based assessment and Instruction

## Circles and Stars

Sophie is playing a game called Circles and Stars. Sophie rolls one number cube and gets the number six. Sophie draws six circles on her paper. Sophie rolls the number cube again and gets the number three. Sophie draws three stars in each of the six circles. Sophie writes 6 X 3 on her paper. Sophie plays the game again. Sophie rolls one number cube and gets the number three. Sophie draws three circles on her paper. Sophie rolls the number cube again and gets the number six. Sophie draws six stars in each of the three circles. Sophie writes 3 x 6 on her paper. Sophie says she got the same total number of stars both times! Is Sophie correct? Show all your mathematical thinking.

### Multiplication Unit

The Multiplication Unit involves identifying a variety of models to represent the process of multiplication in order to learn how to use it to solve problems. Questions to answer may include:

• How do multiplication situations differ from addition situations?
• How do equal-sized groups model multiplication situations in the world outside of the classroom? What real-world examples of equal-sized groups can you think of?
• How do arrays and area models represent multiplication situations in the world outside of the classroom? What real-world examples of arrays can you think of?
• Given a multiplication equation, how can you create a situation to match it?

### Math Concepts and Skills Covered

The student develops and uses strategies for multiplying whole numbers in order to solve problems. The student:

• Finds the total number of objects when equal-sized groups of objects are joined or arranged in arrays up to 10 by 10.
• Represents multiplication facts using a variety of methods.
• Uses a variety of strategies to multiply a two-digit number by a one-digit number. Strategies may include mental math, partial products, and the commutative, associative, and distributive properties.

This task requires the students to know that multiplication involves finding the whole when they know the number of equal parts and the number in each part. Students must also use a variety of models to represent multiplication situations such as equal groups, rectangular arrays and/or equal jumps on a number line.

### Underlying Mathematical Concepts

• Creating multiplication situations to match an expression
• Finding the product when both factors are known
• Number sense to 18
• Commutative Property

### Possible Problem-Solving Strategies

• Model (manipulatives)
• Diagram/Key
• Table
• Tally chart
• Number line
• Array

### Formal Mathematical Language and Symbolic Notation

• Model
• Diagram/Key
• Table
• Tally chart
• Product
• Factor
• Set
• Array
• Row
• Column
• Number line
• Total/Sum
• Dozen
• Greater than (>)/Less than (<)Equivalent/Equal to
• Odd/Even
• 1/2
• Rule
• Variable
• 3 · c = s
• 6 · c = s
• Equation
• Commutative Property
• Expression

### Possible Solutions

Sophie is correct, she did get the same total number of stars both times.

### Possible Connections

Below are some examples of mathematical connections. Your students may discover some that are not on this list.

• Repeat the activity with other rolls of the number cubes.
• 6 is a half dozen.
• 6 threes is 1 1/2 dozen.
• Patterns: Stars +3 or +6, Circles +1.
• When you add equal groups on a number line, you jump over the same number of spaces each time moving to the right, away from zero.
• Extend the number of equal sets of 3 beyond 6.
• Solve more than one way to verify answer.
• Rewrite the story with a new expression.
• Explain how 6 x 3 and 3 x 6 are both 18 but are used differently to represent the situation in the game.
• 6 x 3 is an even number times an odd number which gives you an even product.
• Generalize and prove the rules 3 · c = s and 6 · c = s (key: c is circles, s is stars).

Our teacher-friendly tasks are designed to support both the Common Core and Citywide instructional expectations. GO Math! alignments are also available.
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## Here's What People Are Saying

The Exemplars program is designed to assess students' problem-solving and mathematical-communication skills. It also supports higher-level thinking and extension of mathematical reasoning.

S. Dement
Teacher

Converse, TX

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