Juan is 36 years old, and Jack is 11 years old. In how many years will Juan’s age be twice Jack’s age. I chose this problem because the students can use a diagram, table, guess and check to solve and it only basic operations are needed. Some students may need to coached on what twice means, but other than that, this problem will cause them a little stretch but the solution can be achieved.
Answer: in 14 years, Juan will be twice as old as Jack.
Give the students a timed multiplication test and a timer. Allow them to record their own time to see how fast they can complete the test. You can level the test to meet the needs of all of the learners. For those students who struggle, give them an easier timed test. For the more accelerated student, give them the harder timed test. The purpose of this task is to get them interested in how tests are timed and use this to apply it to their next task.
Any timed multiplication test will do. Students should be able to do 60 problems in a minute. However, the purpose of the activity is to set the stage for the NCTM problem they will be working on in the next section.
To access some timed tests for free, you can go to MathAids.com
Students will be working on the following task from NCTM’s exemplars. The task asks them the following:
On December 1st I did my multiplication facts test in 4 minutes and 20 seconds. On December 2nd I did my multiplication facts test in 4 minutes and 15 seconds. On December 3 I did my multiplication facts test in 4 minutes and 10 seconds. My goal is to do the test in 2 minutes. If the pattern above continues, on what date will I reach my goal?
The goal of this exemplar is to get students thinking about the pattern and the best way to represent it. Students can use a table to help organize their data (SMP 4). Most students will use a chart to solve and then extend the pattern (SMP 2) when the students work it out this way, they will find that on December 29th their time will be 2 minutes. Students that are working to ability will use math representations and math language to communicate their solutions. Those not working to ability will have an error in identifying or extending the pattern or their solution is incomplete.
If students are struggling with this task, you can give them the same task , but after saying the goal is 2 minutes, you could include the following statement “if each day I am able to do the test in 5 seconds less than I did the day before, on what date will I reach my goal?”
This scaffolding can be put into place to help students see that each time you take the test, 5 seconds comes off. Students will also need help in setting up the table. Ask the student what would be a good way to organize this information?
For an extension, the times could be replaced with the following numbers:
4 minutes 20 seconds Dec 1
4 minutes and 14.75 seconds Dec 2
4 minutes and 9.5 seconds Dec 3
The goal is to do the test in 2 minutes. What day will the goal be reached?
The extension has a pattern that is not as accessible. Students will need to have a solid foundation of adding and subtracting decimals to solve this problem. Also, the answer to the more challenging problem can either be December 27th or December 28th. I would accept either answer with justification.
This exemplar will fit in nicely in many different strands of common core. Additionally, the students could graph their results on a coordinate grid to extend their pattern and make a prediction.
In keeping with NCTM's true grading form, I chose to label the students work with Novice, Beginner, Apprentice and Expert. This assignment was worth 10 points. The expert received 10 points. The apprentice received 9 points. The beginner received 8 points and the novice received 7 points. I did not want anyone to fail this assignment so I made sure the lowest grade would only be a "C".
Students that received a rating of "expert" showed a strategy that required them to think about time. They used calculations on their paper.
Students that received a rating of "apprentice" found the pattern and used a table to extend the pattern. They found the solution using the table.
Students that received a rating of "beginner" found the pattern, but made mistakes within their table to cause them to get a wrong solution.
Students that received a rating of "novice" put something down on paper, but it made little sense or was difficult to follow.
See the attached samples of student work.
For the closure of this lesson, I want the students to look at how other people solved this task. I will be doing a HUSUPU to get them working with a partner and go over their solutions. Once the students are in pairs, I’m going to have them share their thoughts on how they came to their answer. I will be listening for mathematical language and representations being used. For example, I want students to say they found the pattern to be 5 seconds less than the time before so I put the information in a chart, extended my pattern and found my solution to be December 29. For students that are working above and beyond, I will be listening for justifiable arguments that December 29th falls in the winter break and that they would need to take this into account when figuring out their solution. Students that are working on the more challenging problem can still participate in this activity. They can explain to their partner what their task was and how they went about solving it. Good learning opportunity for the average student.
After the HUSUPU, allow students to share as whole group. Collect their work as evidence of student learning.