For this warm up, students solve problem number 5 from the division worksheet today: 3 ÷ 1 (With Remainder)-6. Since students did very well with yesterday's word problem and reporting the remainder as the answer to the question being asked, I wanted to give students a different type of division word problem today. I have been thinking about the division assessment I gave to students recently and have spent hours analyzing that data in order to help all my students be successful.
In this warm up problem today, students must perform an additional operation with the remainder in order to arrive at the desired answer. This problem is VERY similar to two world problems on the division assessment that many students scored poorly on.
Many students answered the problem correctly while those that didn't were able to hear explanations from their peers. I do believe my students are going to make great gains by continuing to be presented with problems in which they can apply their skills. I will be anxious to re-assess at a later date.
This video shows a student solving number 5 from the division worksheet.
Students use pattern blocks today to discover fractional relationships. Four key vocabulary words I want students to hear and use today are numerator, denominator, unit fraction and a fraction chain. It is important for students to understand that a single part of an equally divided whole is a unit fraction. Just as we have units of time and units of measurement.
For this lesson, students need a set of pattern blocks. (Only the yellow hexagons, red trapezoids, blue rhombuses, and green triangles are needed. The students do not use the orange triangle or the tan rhombus for this lesson.)
The use of manipulatives is crucial in developing students' understanding of fraction ideas. Manipulatives help students construct mental referents that enable them to perform fraction tasks meaningfully. Therefore, manipulatives should be used at each grade level to introduce all components of the curriculum on fractions. Manipulatives can include these models: fractional parts of circles, Cuisenaire rods, paper-folding activities, and counters
Based off of prior experience and textbooks, the most common regions studied at the elementary grade levels are the rectangle and circle. The "region" represents the "whole," and parts of the region are all congruent. I wanted to expose my students to a variety of shapes and not be limited to the rectangle and circle. I believe it is important that students work with a variety of regions so that they do not think of the region as only "pieces of a pie or pizza." For this reason, pattern blocks are an appropriate tool for work with the region model.
To begin this lesson, I give students 7 minutes to "play" with the pattern blocks. I have found from past experience that students needs a few minutes to explore the manipulatives, build with them, touch each one, and basically play and manipulate them on their own terms before they are ready to use them as a learning tool. I use this website to set a timer that students can see. I have found when they can keep track of their time, they are also more able to focus when "their time" is up.
This photo shows two students building a design with their pattern blocks while the timer is going. This "play" time is important and the students really enjoy it.
After 7 minutes, I direct students to place the yellow hexagon in front of them. I tell students this will represent our "whole." For most of this lesson, I explicitly model how to use the hexagon and other shapes as a region model to represent a whole. I direct most of the lesson today with the expectation that tomorrow, students will direct more of their learning.
I ask students to place triangles onto their hexagon to make a whole. Then I ask how many parts make a whole, emphasizing that this is the denominator. Students respond with "6 parts make a whole." I then ask them questions like:
What fraction is 2 parts of 6? What fraction if 3 parts of 6? How would we write these fractions?
I model how to write 2/6 as a sum of unit fractions: 2/6 = 1/6 + 1/6 I tell students that when there is one part of the equally divided whole, that is a unit fraction. This is an important concept to understand since this is the basis for adding, subtracting and multiplying fractions. For example, when students understand that in the fraction 2/6, 6 is the unit and there are two of them, it builds their understanding of 1/6 + 1/6 = 2/6 and eventually 2 x 1/6 = 2/6.
I repeat the above procedure, modeling, writing, asking questions for many shapes and parts, concentrating on using the terminology numerator, denominator and unit fraction.
At the end of this lesson I ask students what the "bottom number" or denominator means, and what the "top number" or numerator means. I want students to be able to describe that the denominator tells that the whole is divided into that many equal parts and that the numerator is the number of parts or unit fractions in the fraction.
In this lesson students work on:
Common Core State Standard 4.NF.B.3 - Understand a fraction a/b with a > 1 as a sum of fractions 1/b,
4.NF.B.3a -Understand addition and subtraction of fractions as joining and separating parts referring to the same whole, and
4.NF.B.3b - Decompose a fraction into a sum of fractions with the same denominator in more than one way, recording each decomposition by an equation. Justify decompositions, e.g., by using a visual fraction model. Examples: 3/8 = 1/8 + 1/8 + 1/8 ; 3/8 = 1/8 + 2/8 ; 2 1/8 = 1 + 1 + 1/8 = 8/8 + 8/8 + 1/8.
I lead a very brief class discussion and ask students to define numerator, denominator and unit fraction in their own words. Students are still struggling some and having some productive struggle with the terminology and meaning. The class discussion was a reminder to me about how much time students need to develop concepts in order to own that knowledge.