My students have been focusing on using the basic 4 algorithms. I want to take them deeper. I want to see how they communicate and use reasoning skills to determine mathematical patterns.
I invite students to engage in a discussion about their reasoning using appropriate mathematical language. I want them to explain how shapes numbers and patterns are increasing or decreasing I invite students to the carpet. I draw a large function table on the board.
I enter in numbers to see if students can notice the pattern. Some students noticed that all the numbers I entered into the table were odd numbers. So, I take a moment or two to expound on the students response to see exactly how much they know. Does an even number multiplied by 2 results in an even or odd number? Why do you think this? I point out the first two numbers in the chart, and that the function is noted in the right hand column of the chart. I ask students to read the function aloud with me. I want to be sure that students understand that the function in this model is already given. (double the number plus one) I ask student volunteers to write the given problem on the board. The student writes…. 0(0) + 1 = 1 I give other students a chance to ask questions if they need to.
I continue this until students are able to see and explain the function for this table. I point out that other function tables can have different functions; however, they work the same if you keep working on the same pattern. Since all students are feeling pretty comfortable working with function tables I move them a little deeper into the lesson.
Since this concept goes beyond the Fourth Grade Critical Area of Focus to address Analyzing, I allow them time to explore.
In this lesson the following Mathematical Practices will be embedded:
MP.2. Reason abstractly and quantitatively.
MP.4. Model with mathematics.
MP.5. Use appropriate tools strategically.
MP.6. Attend to precision
MP.7. Look for and make use of structure.
MP. 8. Look for and express regularity in repeated reasoning.
Because some students require additional support, I want to have time exploring pattern rules using shapes.
I use stars to create a pattern. I start by placing one star. I say that the pattern is to add 3 more stars each time. Can you create this pattern using your stars? I give them about 3-4 minutes to explore solving on their own. Then I ask them to turn and explain their method to their partner. I want to see if anyone solve this equation differently. I notice some students first thought to multiply. I probe them a bit more to see if they can tell me if the pattern numbers were odd or even. I ask questions like: How do you know? Can you explain? Can you create your own pattern? Did you see a pattern? If so, can you explain?
I continue writing different patterns on the board for students to create using the stars. As students are working I take notes of students who seem to be struggling, so that I can support them throughout the lesson.
In this portion of the lesson I want my students to investigate how and why the given function works. My students seem to like the idea that they can input numbers to get a new total. But, I want them to understand that a pattern is a sequence that repeats the same process over and over. (MP8) As students are working I circle the room just to make sure that they justify the reason for the rules of each function table. Some students notice that all the numbers are odd. I tell students that sometimes the conjunctures we make about a function may not necessarily lead to be true. Some function tables begins with different assumptions, however, the pattern is embedded within the numbers.
Students solve the problems at ease, so I tell them to turn and talk to their neighbor. I want to see if any students came up with different solutions. As students are talking their way through the problem, I probe them a bit. For instance, one student solved the problem correctly; however, he had difficulty explain how and why the given function applied to the table. To assist him in his learning, I ask him what do you notices about the numbers in your table. Are the numbers odd, or even? Why do you think this is? Can you explain why the numbers you entered in the table are all odd/even? I repeat these questions throughout this activity to make sure students focus on the intended purpose of this lesson.
For struggling students, I scaffold them through a bit more than students who appear to be working with ease. I point out that the function of the table is listed on the top right side of the table. I ask them to read it aloud with me. (Double the number plus one) Then, I point to the first number in the chart. (0) Can anyone tell me how to write the function. 0(0) + =1 Right! I ask students to look at the number they entered into the chart. Is that number odd or even? I ask students to write an O for odd and an E for even. One of the students notices that number 1 is an odd number.
Re-teaching odd and even
Some students did understand the difference between odd or even, so I draw a one circle on the board to represent the answer. I note that the circle did not have a partner therefore 1 is an odd number. To make it a little bit clearer I write the number 2 on the board. Then, I draw two large circles on the board to represent the number 2. I draw a large circle around the two circles to show students that the circle now has a partner and none left over, therefore, number 2 is an even number.
When the given time is up, I ask students volunteers to share. As students are sharing their work, I invite other students to ask questions, and to share out if they interpreted the given problem differently. I use students’ responses to determine if students have mastered the given skill, or if additional time is needed.
For struggling students I want to make sure I give them a larger print of a function table. I make sure to repeat the directions, and allow them time to work on their own.
Materials: Independent function tables.docx
Now that students have investigated, discussed, and explained how the function applies to the given table. I want to see if they can fly on their own. I encourage students to really meet the objective of this lesson by demonstrating how and why they got their answers. I explain that just giving the answer is not enough. I encourage students to think about why the function is repeated, and why the numbers they enter are odd or even. I do not expect all students to automatically explain their reasoning with precision. Therefore, I circle the room as they are working to probe them a bit. I want to focus their attention back to the purpose of this lesson. For instance, I may say what the function of this table is. Can you show me how to apply it? Why do you think this rule is repeated? How do you know? Some students tend to stop using the rule, so reminding them help supports their learning. I continue on observing students to see what they are thinking. Some students can quickly complete the table after noticing the pattern. However, some students struggled. I sit with them a bit and ask them to explain whether or not the pattern works. This will help them develop their explanations and critique their reasoning. (MP3) I use their responses to determine if additional time is needed on this objective. If some students finish early, I encourage them to write their experiences in their math journals. I use their responses in their journals to determine if additional time is needed on this objective.
Closing: I ask student to write about their problem solving strategies in their math journal. I do this because students need to know how to explain their answers mathematically.