Students will extend linear patterns with negative rates and negative initial values. Make near and far predictions about the pattern and describe the pattern in words.

The purpose of the Day 3 activities is to reinforce key concepts from Day 1. Students will engage in a math talk, subtracting on a numberline, and will have an opportunity to complete independent purposeful practice. The math talk and purposeful practice serve to develop a deeper understanding of the following big ideas.

• Patterns can be extended because they are repetitive by nature.
• Pattern rules are generalizations about a pattern, and they can be described in words.
• One common use of linear patterns is predicting future events.
• A pattern can be extended to make a prediction.
• For far predictions, calculations are required for efficiency.
• In a growing pattern, the values increase and the rate is positive.
• In a shrinking pattern, the values decrease and the rate is negative.
• In a linear pattern, the values increase at the same rate.
• The rate can be determined by finding the difference between the value of two terms through subtraction.
• The order in which integers are written in a subtraction statement matters because the commutative property does not hold true.
• Constant difference refers to the idea that the difference between two numbers does not change after adding or subtracting the same quantity to both numbers.
• The initial value in a linear pattern is the constant.
• When the initial value is not zero, the relationship between the two variables is not proportional.
• Graphical representations of linear growing patterns appear as straight lines.
• The initial value in a linear pattern is the constant.
• When the initial value is not zero, the relationship between the two variables is not proportional.

Present the following subtraction sentences one at a time. Create a context that encourages students to leverage the subtraction structure of difference. For example, the temperature rose from 5 degrees to 12 degrees. How much warmer is it? The temperature rose from to -9 degrees to -2 degrees, how much warmer is it? Encourage students to use an open number line to defend their answer. Model student thinking on the open number line.

12 – 5
7 – 0
3 – (-4)
-2 – (-9)
-3 – (-10)

Facilitator note: The first four expressions are part of a string. Students will quickly realize that all four expressions have the same difference, they are simply adjusted down the number line by subtracting the same amount from both the minuend and the subtrahend.
The fifth equation is unique. Encourage students to adjust these values along the number line to defend their answer. For example, if I add 10 to both -3 and -10, my equation is 7 – 0.

Show students the following visual math talk prompt and be prepared to pause the video where indicated:

Students will be prompted with:

Construct a subtraction sentence to represent and determine the difference in temperature. Use a number line to defend your answer.

In this set of visual number talks, we give students yet another opportunity to wrestle with integer subtraction using the linear context of temperature.

As we did on Day 3 of this real world math unit, we begin with rising temperature from one positive temperature to another positive temperature resulting in a positive difference. Again, we want to model this using the number line despite most students firmly knowing and understanding the difference is 7 degrees Celsius in order to encourage the use of the tool for more challenging differences coming later in the string.

Show students the following visual math talk prompt and be prepared to pause the video where indicated:

Students will be prompted with:

Construct a subtraction sentence to represent and determine the difference in temperature. Use a number line to defend your answer.

Again, students are given a very low floor opportunity to not only find the difference between a higher temperature and a lower temperature using the number line as a tool to represent their thinking.

It is important to model student thinking with the number line so that they are later able to leverage the number line as a tool for thinking when we begin working with negative temperatures.

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Students will have an opportunity to independently complete practice questions related to linear patterns with negative integers.

As students are working, you might consider using this opportunity to pose purposeful questions and document student thinking for the purpose of formative assessment. 

Pay close attention to the strategies that students are using. 

Are students: 

• Counting?
• Skip counting?
• Using repeated addition?
• Using the relationship between x and y?

While students are working independently, you might also consider strategically pulling students individually or in small groups to offer guided instruction in order to move them along their developmental continuum.

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We suggest collecting this reflection as an additional opportunity to engage in the formative assessment process to inform next steps for individual students as well as how the whole class will proceed.

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