Number Sense Grade 7
Hook 1 7.NS.A.1
Prompt: Explain how Page 1 problems are different than Page 2 problems.
While I've used chips, colored pens and real world contexts to discuss integer operations, I believe vectors on the number line are the most effective way to represent these concepts. A vector's length and direction align to an integer's value and sign. After some discussion, students may articulate the idea that vectors that share the same direction have their lengths added while vectors that aim opposite directions undo one another, or subtract their lengths (in terms of absolute value). These visuals should help students begin to formulate the idea of an algorithm along the lines of "Same signs add and different signs subtract". The importance of absolute value in differences can also be seen in the lengths of the segments and should be interwoven into students' ideas and discussions about integer operations.
Video: Airplane travel (2:05)
Hook 2 7.NS.A.1
Prompt: Which pairs of vectors will add to zero on a number line?
Students have the opportunity to use tools and practice precision while at the same time gaining an understanding of properties of opposites, though they may lack the exact vocabulary. Some students may benefit from a number line and scissors to cut out the given vectors and assign them values on the number line. Others may work in the abstract toward the idea that equal lengths, or values, in opposite directions undo one another or, to be more precise, the sum of opposites is always zero.
Video: Irresistible Force vs. Immovable Object? (0:53)
Hook 3 7.NS.A.1
Prompt: Choose the correct arrow to add to each number line to model the expression. Explain your answer.
Students have the opportunity to consider subtraction of integers through the use of vectors. Applying their prior understanding of the inverse nature of addition and subtraction, students must consider how a given sign, either addition or subtraction, effects the direction of the vector. Some may begin to come to the understanding that subtracting a negative must be the same as adding a positive based on logical reasoning. Of course, they will need time and additional practice to solidify their grasp of this concept but the vectors offer students a concrete tool to reflect upon and discuss the nature of integer subtraction.
Video: Understanding the Stock Market (3:33)
Hook 4 7.NS.A.1
Prompt: Which two equations correctly represent the given scenario.
This hook asks students to consider the application of integer operations in a real-world scenario. Students may be thrown by the fact that all the equations are correct. However, if we consider 32 degrees our starting point, -20 our ending point and -52 our change, only 32 + (-52) = -20 and 32 - 52 = -20 represent this scenario. The order of numbers may seem trivial to students but when we begin applying values to real-world applications the proper representation matters and the discussion of precision is worth having as students will be expected to write variable expressions and equations in the near future.
Video: Cold! (0:47)
Hook 1 7.NS.A.2
Prompt: What should be the sign of the answer if I multiply three negatives? Four negatives? Five negatives?
Some students will be able to discern that a negative times a negative or a negative divided by a negative must be a positive. However, if kids stop there, they will often later forget the proper wording of the rule or apply it to addition or subtraction. This hook puts students' focus on the number of negatives in the problem and asks them to find a pattern. No doubt, some will struggle on their own but a class discussion that introduces multiplication and division of integers by focusing on the number of negative signs in the expression will simplify these algorithms for students and build conceptual understanding of why these algorithms work.
Video: Good Debt vs. Bad Debt (2:10)
Hook 2 7.NS.A.2
Prompt: Place each fraction where it belongs on the number line.
If you work with struggling students, this hook may be an eye opener. Students often learn fractions through the visualization of pizzas or other geometric shapes. However, if this piece-to-whole understanding has not been translated to a number line, students may have a complete lack of understanding of how to relate fractions to other numbers. (I owe a big thanks to the incredible work of the Poincare Institute at Tufts University for first pointing out this discrepancy to me.) So to begin, this hook offers solid formative assessment on your students grasp of fractions. Second, it asks students to consider fractions with negative values. The conversations students have with one another and the ideas they present in class discussion will tell you a lot about the work that needs to be done with rational numbers.
Hook 1 7.NS.A.3
Prompt: Explain the correct answer.
Students seem to have no problem memorizing the order of operations but often struggle to correctly apply the rule. This hook simply reminds them of the importance of this concept and offers insight to why the rule exists in the first place; without it, we would have several different answers to the same problem. If you witness students making simple calculation mistakes, you can highlight organization and encourage students to complete the problem one step at a time, thus creating an inverted triangle of work that results in a single answer.
Video: Order Matters (2:34)