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. Encourage students to use an open number line to defend their answer.

Construct a subtraction sentence to represent the difference in water height from last year to this year. Use a number line to defend your answer.

12 – 8
8 – 12
5 – (-1)
-1 – (5)
-3 – (-7)
-7 – (-3)

Facilitator Note: The big idea here is difference. We are looking at the difference between these two values. However, the order matters. Unlike with addition, subtraction does not follow the commutative property. Is the minuend more or less than the subtrahend?

Minuend − Subtrahend = difference

When the minuend is greater than the subtrahend, the difference is positive. When the minuend is less than the subtrahend, the difference is negative. Seeing these values on the number line should be helpful to solidify this generalization.

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 water height from last year to this year. Use a number line to defend your answer.

This context builds on the work students have already engaged in over the first two days of the Rising Water 6-Day Problem Based Unit on the Make Math Moments website involving a glacier melting and increasing water levels over time.

In this case, we are emerging the idea of subtraction as a difference between the water height this year compared to last year.

In the first prompt in this string of related problems, we provide students with a higher water level this year than last to result in a positive result. In later prompts in this string, we will do the opposite to result in negative differences.

Although students are able to solve this problem however they choose, we will explicitly share a linear model as a tool to represent thinking so we can leverage that same tool once negative differences come into play 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 water height from last year to this year. Use a number line to defend your answer.

In the second visual number talk prompt from this string of related problems, we reverse the water heights so that the water level is decreasing from last year to this year. The result is that the difference will be a negative measure and we can explicitly show this on the number line to again promote this model as a tool for thinking as well as a tool to represent thinking.

While it might seem obvious that the result must be a negative number since the water level has decreased from last year to this year, we are again leveraging the number line to build our confidence with the model to help us when the situation becomes more complex in future problems 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 water height from last year to this year. Use a number line to defend your answer.

Things get more complex when the water level last year was 1 cm below sea level and rose to 5 cm above sea level this year. By leveraging the context and the linear model, students are able to make sense of the difference being 6 cm from last year to this year without the need of mindlessly memorizing rules of subtracting integers.

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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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