Why hello, there teacher friend! I'm back this week to share about the last 4 Standards for Mathematical Practices. If you missed my previous post that included information about the first 4 practice standards, you can find that here.
Let's jump in, shall we?!
Standard 5: Use appropriate tools strategically
Standard 5 is all about students being able to select the most appropriate mathematical tools to be able to successfully complete a math task. Typically when you hear the term math tools one might think of a ruler, protractor, or calculator right? While that isn’t incorrect, the term math tools have a much broader definition when we’re looking at Standard 5. Tools are anything that can support students to perform a task. This could be concrete materials such as base-ten blocks, connecting cubes, counters, a number line, etc. Calculators, paper and pencil, and mental math are also math tools.
Often times there is more than one tool that can be used for a task, but some tools are more efficient than others. This standard is all about students choosing the tools that are MOST efficient in solving their math tasks.
Standard 6: Attend to Precision
This standard in particular is referring to precision in calculations and performance of math tasks as well as precision in communication.
In math, there are times where estimation works. There are other times when precise calculations are vital (balancing a checking account, ordering window blinds to fit within a frame, calculating hourly pay, etc). Initially, precision may be answers expressed in whole numbers, but as our students get older and progress in their understanding of math, that precision is refined with answers expressed as decimals to the tenths, hundredths, or thousandths.
In addition, students must also be precise in other math tasks such as constructing graphs, determining the probability of events, using a ruler, measuring angles, etc. As primary students learn about measurement, they may start with lining up cubes to measure an object. They might notice gaps between the cubes whereas their neighbor doesn’t have any gaps and their measurements differ. As students develop their skills, attention is placed on moving the cubes together to eliminate gaps, and in turn, their measurements become more precise.
When we talk about precision in communication, we expect students to thoroughly describe math ideas, precisely explain how they solve a problem, and give specific examples as they construct their mathematical arguments. In order to do so, students must know the words that express their thoughts. This is where math vocabulary comes into play.
If a student is explaining that they measured “around it” rather than talking about the perimeter of the figure, this student doesn’t quite have the vocabulary to precisely explain their thinking.
How do I get my students to be able to do this?
Model precise communication by using grade-level appropriate vocabulary.
Discuss important math vocabulary and explore the meanings of math words through familiar language, words, pictures, and examples
Expect precise communication and ask students to elaborate on ideas, choose specific words, specify units of measure, and explain symbols they use.
Orchestrate ongoing experiences for students to talk and write about math.
Require students to label units, quantities, and graphs
Expect students to justify labels (e.g., what area is labeled as square units and volume is labeled as cubic units
Model specific and thorough explanations
Allow students opportunities to work with partners to come up with explanations
Standard 7: Look for and make use of structure
Standard 7 focuses on understanding the structure of mathematics and using this understanding to simplify things! For example, when adding up how much your grocery list will cost, you don’t worry about the order in which you add the items. If your students understand math properties, they know that the order in which numbers are added will never change the total. Another example of this standard in real life is when baking. If you can’t find the ¾ measuring cup, you could simply use the ¼ cup three times because you know it’s the same amount or quantity.
This standard is where conceptual knowledge is necessary. Students must truly understand how math works to be able to apply properties and see patterns. The goal of this standard is to get students to see the flexibility of numbers, understand properties, and recognize patterns and functions. Below are a few examples of each.
The flexibility of Numbers: This is the understanding that numbers can be broken apart and put together in a variety of ways. Using this understanding can help students to be more efficient in how they choose to solve problems. This also helps students tremendously with mental math.
6 x 8 is the same as 5 x 8 plus 1 x 8. 6 x 8 is ALSO the same as 6 x 4 + 6 x 4.
53 + 18 could be solved by adding 53 + 20 = 73 - 2 = 71. Here the student understands that it is easier to add 20 to 53 and knows that 18 is 2 less than 20.
Understanding Properties: Understanding the different properties helps students to be able to solve problems more efficiently.
If a student was asked to solve the following problem:
The cafeteria had 14 rows of seats and 8 seats in each row. How many people can be seated?
A student that understands properties can quickly and easily solve this problem in their head. They will immediately break the 14 rows apart into 10 rows and 4 rows. In their head, they would figure out 10 x 8 = 80 and 4 x 8 = 32 and 80 + 32 = 112. This understanding of how distributive property works helps the student simplify the task.
Recognizing Patterns and Functions: When students are able to truly make sense of math, they can look at the numbers to find patterns. Look at the table below.
In this table, we see that as the fractions are being halved, their decimal equivalent is also halved. Using this knowledge students can solve other types of problems.
For example, if you don’t know the decimal representation for 3/8 but we know that 1/4 is .25 and that 1/8 is half of that or .125, you could triple .125 to find that 3/8 equals .375
Standard 8: Look for and express regularity in repeated reasoning.
Standard 8 is all about students seeing and finding patterns and repetition in what they are doing in math. By recognizing the repetition, students are then able to develop shortcuts- like algorithms or formulas to make tasks easier.
In Kindergarten students notice repetition in the counting sequence, always following a 1 to 9 sequence. Once they recognize this, they are then able to continue their counting with “twenty-one, twenty-two, twenty-three….thirty-one, thirty-two, thirty-three….forty-one, forty-two, forty-three…..”.
Standard 8 is where you as a teacher help your students to discover patterns and make sense of them to gain a deeper understanding of math as opposed to just telling them. Letting students discover on their own will make the learning stick.
How do I get my students to be able to do this?
Set up learning opportunities for students to gather data and observe repetitions to find shortcuts.
Ask students to explain shortcuts
Frequently ask “What do you notice?” “Do you see any patterns?” “Have we seen this before?”
Pose problems that draw attention to repetition
Ask students to think about how new problems are like previously solved problems
Ask students to use familiar problems as a way to decide on an appropriate strategy
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