##
* *Reflection: Staircase of Complexity
Patterns in Addition - Section 2: Warmup

We had an opportunity here to dig a little deeper and not stop at a final answer. The real value in sharing multiple solutions or multiple representations is in comparing them to each other to uncover the structure and patterns in the math. This is what helps students understand foundational concepts and develop number sense.

After several of my students shared their solutions to the first triangle I asked them to discuss **"what was the same about all of them" or "what was true about all of them".** Several responses included the idea that, in order to result in a sum of zero the two bottom numbers had to be "the same, but different signs". Thinking about the numbers in this generalized way helps my students think about the number relationships and develop number sense.

I then asked another series of questions to get them to clarify by using more precise language:

**"Why does it make sense that being 'the same with different signs' will have a sum that is equal to zero?"****"what would that look like on a number line?"****"Is there another way to represent that?"****"do we have to use numbers to show they are the same?"**

By the end they were relating the 'sameness' and 'oppositeness' to distance on a number line, saying "the two numbers have to be the same distance on the number line in opposite directions" (absolute value).

This precision made it easier to talk about the third number in the triangle. My students said the number had to be positive, so I followed up with questions like:

**"How do you know?" or "what makes you say that?"****"Will any positive number work?"****"Which positive numbers won't work?" and "why not?"****"how does the top number have to relate to the other two?"**

After this conversation my students were able to tell me that this positive number had to be a greater distance on the number line from zero than the negative number.

**Taking the time to discuss the puzzles and generalize will make the following explorations move faster. Students will notice the patterns more readily and be able to discuss their ideas more clearly. They may even begin asking themselves and each other questions to help them dig deeper.**

*Staircase of Complexity: Digging deeper*

# Patterns in Addition

Lesson 9 of 24

## Objective: SWBAT manipulate an integer sum to make it positive, negative, or zero in order discover the pattern in the number relationships.

*54 minutes*

The purpose of this lesson is to give students the opportunity to "play" with integers to produce a sum that is either positive, negative, or zero. Students should become familiar with the relationships required to produce such a sum. I want them to notice that the sum of two integers will be negative if more negatives are being added than positives, or if only negatives are being added and vice versa for a positive sum, and that adding opposite numbers results in zero. This lesson puts students in the "driver's seat" by having them create the problems that will result in a specified sum.

*expand content*

#### Warmup

*15 min*

In this warmup secret number patterns students are given three "secret number" triangles telling them that a "secret number" is hidden at each vertex of the triangles. The number on the sides of the triangles are the sums of the two numbers each end. In these triangles students are not given any of the "secret numbers" and are not given the numbers on the sides except whether they are positive, negative, or zero. Students are supposed to find "secret numbers" which will result in sums that match the given criteria. Answers may vary, but so long as the "secret numbers" create a sum that matches the criteria it is correct.

#### Resources

*expand content*

#### Exploration

*35 min*

The warmup is replaced with positive negative zero sum set of two "secret triangles" with nothing given but the sign of the "secret number" and students are asked how they know if the sums will be positive, negative, or zero.

One triangle has all negative "secret numbers" and the other has two negative and one positive.

Students discuss in their math family groups what the options are for each side of the triangles. This is a great way to generate argumentation by having them defend their claim with an example or an explanation or by having them challenge the claims of others by requiring them to use evidence. I ask them to look for all the options, so they don't just stop at one possibility. If they say it can be negative, I ask if it could also be possible to get a positive or zero sum.

For the first triangle, with all negative "secret numbers" the sums can only be negative, but for the other triangle, two of the sides could be either positve, negative, or zero, while one can only be negative. I have students discuss and explain the relationship between the numbers that produces each given possibility.

The next patterns in addition gives two of the "secret numbers", 2 at the top and -6 in the lower right (which makes the right side of the triangle a sum of 4) and I change the criteria for the sums on the other two sides and ask them what the third "secret number" (n) could be in each case.

First I ask what value of "n" would make the bottom side of the triangle a sum of zero (+6) and what would that make the left side (+8). Then I ask what value of "n"would make the left side of the triangle a sum of zero (-2) and what would that make the bottom side (-8). This gives them a feel for the type of questions I'm going to ask them. I ask what "n" could be that would give both remaining sides a negative sum. They should come up with several possibilities which we try and I write down (n=-3, -4, -5,etc) I ask them why it can't be -2 or a positive number. I ask them how they could describe ALL the possible values. I am looking for something like "any negative number between -3 and -infinity" or "all the numbers less than -2 or less than or equal to -3". If they say "greater than", I show them on a number line that they are less.

We go through the same process when I ask them what value of "n" could make the bottom side a negative sum and the left side a positive sum, both positive. I ask if there is a way to make the left side negative and the bottom side positive (no). The key is making them explain why and why not. If they are really picking this up I like to extend it a little by showing them the possibilities on a number line and in an inequality (-2 < n < +6).

*expand content*

#### Homework

*4 min*

I let them get started on their homework patterns addition for the remainder of class. I tell that for the first two, which are "secret number" triangles, they only have to come up with one solution, but for extra credit they can find all the possible solutions.

#### Resources

*expand content*

##### Similar Lessons

Environment: Urban

Environment: Urban

Environment: Suburban

- UNIT 1: Order of operations & Number properties
- UNIT 2: Writing expressions
- UNIT 3: Equivalent Expressions
- UNIT 4: Operations with Integers
- UNIT 5: Writing and comparing ratios
- UNIT 6: Proportionality on a graph
- UNIT 7: Percent proportions
- UNIT 8: Exploring Rational Numbers
- UNIT 9: Exploring Surface Area
- UNIT 10: Exploring Area & Perimeter

- LESSON 1: Cooking with The Mathmaster Chef (Day 1 of 4)
- LESSON 2: Cooking with Mathmaster Chef (Day 2 of 4)
- LESSON 3: Cooking with Mathmaster Chef (Day 3 of 4)
- LESSON 4: Cooking with Mathmaster Chef (Day 4 of 4)
- LESSON 5: Which Way Do We Go?
- LESSON 6: Intervention Day
- LESSON 7: The Three Little Bears
- LESSON 8: Secret Numbers
- LESSON 9: Patterns in Addition
- LESSON 10: How Do You Know?
- LESSON 11: Secret Number Sub Plan
- LESSON 12: Patterns in Subtraction Sub Plan
- LESSON 13: Patterns in Subtraction (Day 2 of 2)
- LESSON 14: Patterns in Subtraction
- LESSON 15: Equivalent Expressions
- LESSON 16: Matching Equivalent Expressions
- LESSON 17: Magic Witch Hats
- LESSON 18: Is it Postive or Negative?
- LESSON 19: Integer Addition & Subtraction Assessment
- LESSON 20: Multiplying with Mathmaster Chef (Day 1 of 2)
- LESSON 21: Multiplying with Mathmaster Chef (Day 2 of 2)
- LESSON 22: Integer Product Game
- LESSON 23: Patterns in Mutliplication and Division
- LESSON 24: Integer multiplication & division assessment