Lines and Patterns: Difference, Change, and Multiplication

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Objective

SWBAT solve word problems about difference, change and multiplication by using number lines and patterns.

Big Idea

Students will explore difference and change problems in one mini lesson and discover the rules of multiplication in a second mini lesson

Do Now

6 minutes

Students enter the room silently to begin their Do Now which consists of two addition word problems whose chip models and number line models must be drawn. Directions on the SMARTBoard instruct students to raise their hand when they have finished so that they can draw their representation and give their answers on the white board in the room.

When we review the chip models we talk about the number of “blue” (negative) and the number of “red” (positive) chips students should visualize. “Are there more “blues” or more “reds”? What will be the sign of the answer?” or a higher level of questioning, “Are there more “blues” or more “reds”? What does this mean about the sign of the answer?

When we review the number lines we talk about the direction implicated by the sign of the integer. “Where do we always start? In which direction and how many units should I move?”

I answer questions for the remaining amount of time and we transition to class notes.

Class Notes + Mini Lessons

15 minutes

These notes are broken up into two mini lessons: the first is about subtraction word problems which use the words difference and change and the second will guide students through patterns to discover the rules of multiplying integers.

Students will receive two vocabulary cards for the words difference and change. They are instructed to copy the definition on their cards on the blanks under “notes”. Then, I ask students to talk about the difference between the two words. How are their definitions similar? How are they different? The target discussion is to notice that difference will always be positive because it is the space between two units, whereas change includes a positive or negative sign depending on the direction the change is occurring.  The same word problem is presented in the notes twice with a different question asked. A representation of each word problem should be drawn on the vertical number line. The first problem reads, “The temperature in St. Cloud, Minnesota, was 7°F (Fahrenheit) on January 27th and –4°F on January 28th.  What is the difference between these two temperatures?”.  What is the difference between these two temperatures?” A common mistake in these types of word problems is the memorization that “difference means subtract”. In this example, students may subtract 7 – 4 = 3 and falsely report the difference between these two integers as 3. The number line representation aids students' understanding of the concept of difference as the space between two units, instead of just memorizing cues such as “different means subtract.” The correct answer is 11 degrees.

 

The second word problem reads, “The temperature in St. Cloud, Minnesota, was 7°F (Fahrenheit) on January 27th and –4°F on January 28th. What is the change in temperature?” In this example the answer will be –11 degrees because the temperature is going from a higher number to a lower number. Noting the direction in which the change is occurring on the number line will help students understand this concept and how if differs from the previous word.

 

The multiplication mini lesson will task students to use MP8 by applying multiplication facts and patterns to discover the rules of multiplication. First, students will fill out the first 5 multiplication facts in the table on the left. Then, they will look at the right-hand column to find and complete the pattern reading down that column (“subtract/take away 4”). By filling out this pattern, students will discover the result when multiplying a positive with a negative.

 

After the first table has been filled out, I will use the number line to demonstrate the first multiplication fact in the right table. I ask students to read the math fact, “negative two times 4” and explain what this means? If they need an example, I can offer the following: “2 times four means add 2 four times”. Since  -2 x 4 says add -2 four times, this also means go left 2 units four times. Display this on the number line and ask students to use your example to find the answer to the next 4 multiplication facts. Students then revisit the vertical pattern in the right-hand column to see “add 4”. This will reflect the rules “a negative times a negative equals a positive”… I show the Stand and Deliver movie clip if there’s time. This clip ends with Jaime Escalante asking "why?". Having students answer this question after the activity is a great way to check for understanding.

Task - Paired Partner Work

15 minutes

Students work in pairs to solve word problems in each part of the class work. I use booths in my class classroom to motivate urgency during this work period.  If students complete the first five problems correctly and raise their hand to let me know they are finished, they earn a spot in a booth. There are only two students allowed per booth. Students may also ask to “chelp”.  Once there are 7 minutes left in class, all students are asked back to their seats.

While they are working in pairs, students are expected to enter their answers into clickers. These results will be used tomorrow when re-seating students for a station activity. A common mistake I look out for in this task is drawing of number lines. Problems that do not include a number line, or whose directions do not state that students must draw a number line are sometimes answered incorrectly , especially when asked to find the difference between a positive and a negative. Students whom I notice not drawing number lines will be asked to do so, whether the problem is correct or not.

Closing

10 minutes

I review a selection of the questions most erred by students during paired partner work, data I collected using clickers. I review questions one of two ways: by writing the steps and final answer on the board and asking students to identify the step they either did not understand or the step they completed incorrectly, or I ask a chelper to explain the solution on the board while also sharing common misunderstandings they noted while they were chelping. Students are expected to fix any problems we reviewed together and complete anything they did not finish for homework. They are told that this assignment will be graded.