Daily/Weekly Instructional Guides

Content Area/Subject: Math / Grade: 7th Grade / Day(s) of Instruction: Day 17
Date:
SCoS Objective(s):
7.NS.1a Describe situations in which opposite quantities combine to make zero.
7.NS.1d Apply properties of operations as strategies to add and subtract rational numbers.
7.EE.1 Apply operations to add, subtract, factor, and expand linear expressions with rational coefficients.
7.EE.2 Understand that rewriting an expression in different, yet equivalent, forms in a problem can show how the quantities in it are related.
7.EE.3 Solve multi-step real-life and mathematical problems posed with positive and negative rational numbers in any form (whole numbers, fractions, and decimals). Apply properties of operations to calculate with numbers in any form; convert between forms as appropriate; and assess the reasonableness of answers using mental computation and estimation strategies.
MP.1 Make sense of problems and persevere in solving them.
MP.4 Model with mathematics.
MP.6 Attend to precision.
National Objective(s):
Learner Objective(s): As a result in learning, students should be able to… use the Hands-On Equations Learning System to introduce negative integers and the addition and subtraction of integers. Students will also learn another legal move to solve equations that involves adding the same amount to each side of an equation to help simplify sides.
Language Objective(s): / Allocated Time for Instruction:45 to 90 minutes depending on school
Connections to EOG/EOC/Assessment:
Use the Hands-On Equations Learning System to solve:
4x + (-x) = 15
A. x = 3
B. x = 4
C. x = 5
D. x = 15
The answer is C.
Topic(s):
Use the Hands-On Equations Learning System to:
  • Introduce the green number cubes (negative integers).
  • Introduce a legal move that allows students to solve equations by adding the same value number cube to both sides of a balanced system.

Prerequisite skills/knowledge:
  • Ability to substitute to determine whether a given number makes an equation true. (6.EE.5)
  • Understand that variables represent numbers (6.EE.6)
  • Ability to set-up and solve one-step equations (6.EE.7)
  • Ability to add, subtract, multiply, and divide positive and negative integers.
  • Understand the idea of opposites (7.NS.1b)

Real World importance of objective(s) – Essential Questions
Why do students need to know or be able to do this?
  • How do you solve equations that have negative numbers involved?
  • What are some real-world problems that can be solved using equations? Do any real-world examples involve negative variables? Integers?

Definitions of Critical Vocabulary and Underlying Concepts
rational numbers / a number expressible in the form a/b or –a/b for some fraction a/b. The rational numbers include the integers.
integers / a number expressible in the form a or –a for some whole number a.
constant / a number that does not change.
expression / a mathematical phrase that contains operations, numbers, and/or variables.
evaluate / To find the value of a numerical or algebraic expression.
equivalent expressions / expressions having the same value.
equation / a mathematical sentence that shows that two expressions are equivalent.
additive inverses / two numbers whose sum is 0 are additive inverses of one another.
additive identity property of zero / The property that states the sum of zero and any number is that
number.
addition property of opposites / the property that states that the sum of a number and its opposite equals zero.
subtraction property of equality / the property that states that if you subtract the same number from both sides of an equation, the new equation will have the same solution.
distributive property / the property that states if you multiply a sum by a number, you will getthe same result if you multiply each addend by that number and then add the products.
Introduction/Focus/Anticipatory Activities/Guiding questions/Connections to what students already know through learning, culture, or experience:
  • Ask students to figure out the answer to the following equation through guess and check:
  • _____ + (-2) = 10
  • What would you fill in the blank with to make a true statement? What integer rules did you use?
  • Materials that each student will need:
  • Eight Blue Pawns
  • Eight White Pawns
  • Four red number cubes, numbered 0-5 or 5-10
  • Four white number cubes, numbered 0-5 or 5-10
  • A laminated balance sheet

Initial Instructional Strategies (research-based teaching strategies):
A. Direct Instruction – What the Teacher does:
Today’s lesson utilizes the Hands-On Equations Learning System.
  • Things to know when using resources provided by Hands-On Equations:
-Seventh grade objectives do not cover solving equations with variables on both sides, which is why many lessons in Level II and III will be skipped. It is critical in any of the lessons that are utilized that you are selective in picking problems that do not contain variables on both sides.
-Please always refer to the county-provided Hands-On Equations Learning System resources that are located at your school for additional examples, comments, consistent language, and teaching strategies. Examples provided in this lesson are original examples and are not found in the Hands-On Equations resources.
-For today, students should not be required to write down any steps, other than their checks. They will need a pencil to show how to solve pictorially.
Source: Henry Borenson, Ed.D. The Hands-On Equations Learning System. Pennsylvania: Berenson and Associates, Inc., 2008. Print.
  • All students will need a copy of the Hands-On Equations Student Packet (Same Packet as day 15)

  • All teachers will need a copy of the Hands-On-Equations Teacher Packet (Same Packet as day 15)

Topic 1: Using Level III, Lesson 17, students will work with adding and subtracting positive and negative whole numbers using the number cubes. Although they will have already completed the integers unit and should have a solid foundation with these concepts, it is critical that they understand how to use the number cubes to complete addition and subtraction. Students need to know that the green number cubes represent negatives, so a 3 red-number cube (3) and a 3 green number cube (-3) are additive inverses (negatives) and their sum will be 0.
  • If necessary, use opposite number cubes to demonstrate that these are additive inverses.
Direct Instruction Examples:
  • Example : (-4) + (-5) =
  • To set-up, students should put a 4 green-number cube and a 5 green-number cube on the left side. Walk around to ensure that students have set this up properly.
  • Students already learned in previous lessons that they could combine red-number cubes into the sum of the numbers in the same color number cube. They should extend this information to this example and see that a 4 green-number cube and a 5 green-number cube is the same thing as a 9 green-number cube.
  • So (-4) + (-5) = -9
  • Example : (-4) + 6 =
  • Students will probably be able to answer this easily due to the integers unit, but it is helpful for them to see how to utilize the number cubes. Ask them to break the 6 red-number cube into two number cubes, one being a 4 and the other a 2.
  • Additive inverses (opposites) allows them to see that the 4 red-number cube and then 4 green-number cube will cancel.
  • (-4) + 4 + 2. They can remove this pair of opposites.
  • They will be left with 2 (2 red-number cube).
  • Example : (-4) – 7 =
  • Again, students will probably be able to apply rules that they learned in the integers unit to solve this, but encourage them to demonstrate the rules they know using the number cubes.
  • To set-up, students should put a 4 green-number cube on their balance board. They aren’t able to take a 7 red-number cube away because it does not exist yet, therefore, they should hopefully begin discussing what they can do using rules that they’ve learned in Level II.
  • Students should remember their ability to add a convenient zero pair in the set-up process (not considered a LEGAL MOVE since it is completed in the set-up) to add both a positive 7 and negative 7 (zero pair, sum is 0) to the left hand side. They can then take away the 7 red-number cube.
  • This will leave a 4-green number cube and a 7-green number cube on the balance. So, (-4) – 7 = -11

Expression: / Solution/Check:
Example 1:
Lesson 17 / -6 – (-4) / -2
Example 2:
Lesson 17 / 12 – (-3) / 15
Example 3:
Lesson 17 / -5 – (-9) / 4
(add convenient zero pair of -4 and 4 so that you have -9 to take off during set-up)
Topic 2: Using Level III, Lesson 18, students will learn another a legal move where they can add the same number cube to both sides of the balance to cancel out constants next to it and isolate the variable.
Direct Instruction Examples:
  • Example : x + (-4) = 6
  • To set-up for this example, students will put a blue pawn and a 4 green-number cube on the left and a 6 red-number cube on the right.
  • Remind students that the goal is to isolate the variable. Ask students if they can think of any ways to get rid of the 4 green-number cube (-4). Once students realize that they can cancel it out with a 4 red-number cube, the discussion should proceed into how to remain balanced. In the 6th grade, students should have learned that to solve equations, what you do to one side you must do to the other. So, if they add a 4 red-number cube to one side of the balance scale, they must also add it to the other. This is a LEGAL MOVE.
  • Their balance board should now look like:
  • x + (-4) + (4) = 6 + 4
  • The (-4) and (4) are opposites and can be taken away from the left side since their sum is zero.
  • So, x = 10

Equation: / Solution/Check:
Example 4:
Lesson 18 / x + 5 = -9
(put a 5 green-number cube on both sides to cancel out the 5 red-number cube on the left) / x = -14
Example 5:
Lesson 18 / x + 6 = 2
(put a 6 green-number cube on both sides to cancel out the 6 red-number cube on the left. Students will then need to use their knowledge from lesson 17 to simplify a 2 red-number cube and a 6 green-number cube.) / x = -4
  • Example : 3x + 8 = -4
  • To set-up for this example, students will put three blue pawns and a 8 red-number on the left of the balance scale. They will put a 4 green-number cube on the right.
  • To isolate the variable, the 8 must be removed first by giving both sides a 8 green-number cube. The 8-red and 8 green-number cubes can be removed from the left-side since they are additive inverses and their sum is zero.
  • 3x + 8 + (-8) = -4 + (-8)
  • 3x = -12
  • x = -4

Equation: / Solution/Check:
Example 6:
Lesson 18 / 4x + (-5) = -1
(put a 5 red-number cube on both sides to cancel out the 5 green-number cube on the left) / x = 1
Example 7:
Lesson 18 / 2x + (-4) = 10
(put a 4 red-number cube on both sides to cancel out the 4 green-number cube on the left. / x = 7
Topic 3: Review how to move from the pawns and number cubes to a pictorial representation using paper and pencil. Use lesson 25 for additional assistance, but know that lesson 25 will contain advanced problems that have not been introduced to students. Basic components of pictorial representations as recommended by Dr. Borenson:
-Draw the balance by using one long horizontal line for the scale and a short vertical line that divides the scale in half.
-Shaded triangles will represent the blue pawns.
-Unshaded triangles will represent the white pawns.
-Boxed numbers represent the red (positive) number cubes.
-Circled numbers represent the green (negative) number cubes.
-Students will circle pairs of additive inverses of triangles that they want to remove from a side and use arrows pointing away from the scale to reference the removal.
-Students will cross of boxes/circles (number cubes) when they are eliminated or changed on either side of the scale. They will rewrite a new number cube if a constant is left on one side. (The program uses arrows to show when you take pawns from both sides. Since we are refraining from using variables on both sides in these lessons that may not be necessary to show).
Direct Instruction Examples:
  • Example : 2x + (-4) = 6
  • To set-up, students will draw 2 shaded triangles and a circled 4 on the left side of their drawn scale balance. They will put a boxed 6-number cube on the right.
  • They will then add a 4-boxed number cube on both sides (legal move). Typically, the newly added pieces to both sides are drawn in a row above the original set-up picture.
  • Students should cross off the boxed 4-number cube and the circled 4-number cube (both on the left). They can combine the boxed 6-number cube and the boxed 4-number cube to a boxed 10-number cube if they wish, but it isn’t necessary in this example.
  • Then will then be left with 2x = 10. So, x = 5
  • Check: 5 + 5 + (-4) = 6
  • 6 = 6

Equation: / Solution/
Check:
Example 8: / 6x + 10 = 4
-Set-up: 6 shaded triangles and a boxed 10 | boxed 4
-6 shaded triangles, a boxed 10, a circled 10 | boxed 4, a circled 10
-cross of boxed 10 and circled 10 on the left.
You are left with:
6 shaded triangles | boxed 4, a circled 10
6 shaded triangles | Circled 6
x = -1 / x = -1
**Challenge- Use examples with white pawn.
B. Questions to Promote Higher Level and Critical Thinking (i.e., Socratic):
  • Does x always have to be a positive number?
  • If x is a negative number, then what would –x equal?
  • x and –x are opposites with a sum of 0. What is their quotient?

Instructional Resources (specific to each objective):
  • Primary
  • The Hands-On Learning System by Henry Borenson, Ed.D
  • Supplemental
  • Holt textbook- Selected sections from Chapters 2, 3, and 11.

Guided Practice Strategies or Activities:
  • Guided Practice examples and solutions are provided after each direct instruction example. You can have students complete these after each example OR wait until the end to complete them all at once. Solutions have been provided in the direct instruction section.
  • Hands-On Equations Practice & Homework Set

Formative Assessment (Include context and accuracy percentage)
Informal - Skill/Knowledge Acquisition (Ex. Teacher monitor, analyze student questions)
  • Peer Assessment- Each student had the opportunity to complete the same problems and assist students when there was a lack of understanding. Have students peer assess using a few guiding sentence fragments, such as:
  • You did these really well…
  • You could have…
  • Next time you need to focus on…
Formal - Skill/Knowledge Mastery(ex. Open-ended quiz, prompted written response)
  • Transfer and Apply
  • On an index card, students write down concepts learned from the class on one side; on the other side, they provide an application of each concept (you may wish to provide an example that they complete.

Differentiated Instructional Strategies/Modifications:
  • Students can use the National Library of Virtual Manipulatives for another way to assist them in completing equations.
  • Copies of all notes and classwork examples.
  • Students can check evaluated expressions (equations) using the TI-73

Re-teaching Strategies or Activities:
  • Students should use the Hands-On materials to assist them through the homework questions.

Enrichment:
  • Students should complete the class work/homework worksheet and provide reasons (vocabulary words) for being able to complete each move.
  • Write it out! Instead of representing solutions pictorially, have students write what they would do solve the problems. Have them include properties used when solving for an additional challenge.

Independent Practice: (ex. WebQuest, textbook problem, interview, etc…)
  • Additional Examples from Classwork Sheets provided with each class set of the Hands-On Equations Learning System. These examples will not have a set-up that involves variables on both sides:
  • Lesson 17: #1, 2, 3, 4
  • Lesson 18: #1, 2, 3, 4, 10

Comments /Notes:

C&I/MS Math/Fall 2011