Standard 7: Look for and make use of structure

The Standard:
Mathematically proficient students look closely to discern a pattern or structure. Young students, for example, might notice that three and seven more is the same amount as seven and three more, or they may sort a collection of shapes according to how many sides the shapes have. Later, students will see 7 × 8 equals the well remembered 7 × 5 + 7 × 3, in preparation for learning about the distributive property. In the expression x2 + 9x + 14, older students can see the 14 as 2 × 7 and the 9 as 2 + 7. They recognize the significance of an existing line in a geometric figure and can use the strategy of drawing an auxiliary line for solving problems. They also can step back for an overview and shift perspective. They can see complicated things, such as some algebraic expressions, as single objects or as being composed of several objects. For example, they can see 5 – 3(x – y)2 as 5 minus a positive number times a square and use that to realize that its value cannot be more than 5 for any real numbers x and y.

Classroom Observations:
Teachers who are developing students’ capacity to "look for and make use of structure" help learners identify and evaluate efficient strategies for solution. An early childhood teacher might help students identify why using "counting on" is preferable to counting each addend by one, or why multiplication or division can be preferable to repeated addition or subtraction. A middle childhood teacher might help his students discern patterns in a function table to "guess my rule." A teacher of adolescents and young adults might focus on exploring geometric processes through patterns and proof. Visit the video excerpts below to view multiple examples of teachers engaging students in identifying and making use of mathematical structure.

## Connections to Classroom Practices

Mathematically proficient students look closely to discern a pattern or structure. Young students, for example, might notice that three and seven more is the same amount as seven and three more, or they may sort a collection of shapes according to how many sides the shapes have.

Liz O'Neill works with her first grade students engaging them in composing and decomposing numbers within twenty, focusing on ways that numbers combine to produce other numbers. She begins by having her students rename a "target" number in as many ways as possible in their math message book. They are then given a 2 second quick look at 3 ten frames and asked to determine that number (23) mentally. Using sentence frames, students share with their partner what number they saw and how they saw it. A variety of ways were discussed as a whole group after everyone had a chance to share with their partner. The main lesson activity is a game called "How Many are Hiding?" Student pairs were given a bag with 10 cubes, a paper plate, and the "How Many Are Hiding Recording Sheet". In addition, sentence frames were posted on the board so students could produce academic language using structured student talk and convince their partners with oral justification. One partner takes some of the cubes and "hides" them under the plate. The remaining are placed on the top. The second partner uses sentence frames to answer the questions "What number do you see?", "How many are hiding?", "How do you know __ are hiding"? In addition, the answers are recorded. Roles are then reversed. The partner game gives students practice in composing and decomposing numbers within ten.

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Mathematically proficient students look closely to discern a pattern or structure. Young students, for example, might notice that three and seven more is the same amount as seven and three more, or they may sort a collection of shapes according to how many sides the shapes have.

Becca Sherman works with 4th grade students in a “number talk” to connect the basic components of the Singapore Bar Model with students’ original thinking, thus front-loading students with several applications of the Bar Model as a representation of equal parts. In the exploration problem the words “three times” becomes a division problem or a missing factor problem. The intermediary step of drawing a “math picture” or model of the problem, poses a challenge for many students who have limited exposure to models.

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See this video in the context of an entire lesson.

Mathematically proficient students look closely to discern a pattern or structure. Young students, for example, might notice that three and seven more is the same amount as seven and three more, or they may sort a collection of shapes according to how many sides the shapes have. Later, students will see 7 × 8 equals the well remembered 7 × 5 + 7 × 3, in preparation for learning about the distributive property.

In the closure section of a number talk, Fran Dickinson works with his 5th/6th grade students to identify many different ways of representing the rule: x3 – 3, times 3 minus 3, 3x – 3. The students discuss the rule and the best way to represent it, making connections to their mathematics textbook in their conversations. This clip is also indicative of standard 3 (construct viable arguments and critique the reasoning of others), standard 6 (attend to precision), and standard 8 (look for and express regularity in repeated reasoning).

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See this video in the context of an entire lesson.

Mathematically proficient students look closely to discern a pattern or structure.

Cathy Humphreys leads an extended exploration of a proof of the properties of quadrilaterals, helping students learn to investigate, formulate, conjecture, justify, and ultimately prove mathematical theorems. In these clips, students engage in the first of two block-length explorations of their proofs. Humphreys observes, “The square, rectangle, and rhombus appeared to be the most straightforward for the students. Mathematically, if two of the diagonals form right angles, then at least a pair of sides of the quadrilateral will be equal in length. If the diagonals intersect at the midpoint of both diagonals, then the figure formed will be some parallelogram. In order for two diagonals to form a non-isosceles trapezoid, the following relationships must hold true: If AB is one diagonal and DE is the other diagonal, then trapezoid ADBE is formed only if the diagonals intersect at point P, which is not the midpoint, and AP/PB = DP/PE. This relationship was quite difficult for the students to investigate and conclude. The students did not choose to measure the diagonals with rulers, and therefore did not pick up on the proportional aspects of the diagonals in a non-isosceles trapezoid.” This clip is also indicative of standard 1 (make sense of problems and persevere in solving them), standard 3 (construct viable arguments and critique the reasoning of others), and standard 6 (attend to precision).

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See this video in the context of an entire lesson (Parts A-D).

Mathematically proficient students look closely to discern a pattern or structure.

In the closing of the group work on the first day, Humphreys refers her students to the idea of “mathematical friends.” This notion came from Thinking Mathematically by Burton and Mason, a book about mathematical problem solving in which the authors talk about a hierarchy of certainty when trying to write a convincing argument. Convince yourself (the easiest), convince a [mathematical] friend, and finally, convince a skeptic. Developing a skeptical mindset and not jumping to conclusions too quickly is another hallmark of good mathematical thinking. Humphreys asks to meet with the students who are playing the role of “facilitators” in their groups to ensure that the Burton and Mason argumentation structure is followed. This clip is also indicative of standard 3 (construct viable arguments and critique the reasoning of others), standard 6 (attend to precision), and standard 8 (look for and express regularity in repeated reasoning).

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See this video in the context of an entire lesson.