Teaching Strategies
This unit should take approximately 3-4 weeks to teach. In this section I will go over ideas for daily activities and a general outline of the major concepts and how they break down. I want to encourage all teachers to strive for depth at each stage of the unit. Do not be afraid to delve into an aspect of the unit for a greater time than prescribed. This unit should be contextualized as much and as often as possible. I have used the number line throughout this unit to attempt to give the study of fractions a context in linear measurement. I have done this because I believe our students see fractions as division problems or part to whole relationships and not as rational numbers with value.
Number Talks
Throughout the year, I give number talks at least 3 days per week. A number talk is a 5 to 10 minute mental math problem that highlights the different ways our children solve problems. I will typically use these talks to reinforce foundational concepts such as addition, subtraction, multiplication and division. A problem like 327 + 468, can elicit some valuable information for the children and for that matter, for yourself. Students will solve these problems in multiple ways: some may simplify the numbers: 325 + 475 = 800 then subtract 7 and add 2 giving you 795. Some students may deconstruct the numbers: 400 + 300 + 20 + 70 + 7 + 8. Other students will gravitate towards what they feel comfortable with and stack the numbers and attempt to add in their heads.We need to understand how our children are approaching these "basic operations". Many students cannot escape traditional operations for addition, subtraction, multiplication or division. Seeing others break down numbers, use simple numbers or playing with place value will help many of those students feel more secure when experimenting with numbers.
I have found number talks to be most effective when working with basic operations. Number talks work well when we take the time to ask questions like: How did you think about that? Can you explain the process you used? How did you figure it out? Did anyone get a different answer? Giving children a chance to share their methods can be a powerful motivator for many students. Opening children's eyes to new methods of solving basic operations that they have not considered yet can be a game changer. Kathy Richardson's book, Number Talks, is a wonderful resource for this great daily ritual.
Addition, Subtraction, Multiplication and Division of Whole Numbers
Placing Cuisenaire rods and cubes end to end as a representation of addition and length can be very effective. I realize this will be a very simple section of the unit for our 5 th, 6 th and 7 th graders, but if we take the time to develop the process of concatenation of lengths, the connections when working with addition and multiplication of fractions will go along way in assisting with student understanding. Lesson #1 below, is an example of a possible direction one could take when opening up this unit.
Subtraction is a comparison of the difference of 2 rods. Laying rods next to each other, the greater rod on top, will give a visual reminder of subtractions as difference. Subtraction practice with rods, cubes and number lines should follow or be combined with the addition and placement lessons (lesson #1). Re-establishing the inverse connection between addition and subtraction is a strong step in building the foundations we are attempting to work on at the beginning of this unit.
Multiplication is a process of repetitive addition. If students practice concatenating lengths of the same unit, (i.e. 3 unit Cuisenaire rods) for a given number of times (i.e. 5) they will have a hands on model of the problem 3x5 and the operation of multiplication itself. This is modeled with rods and using a number line below.
Division would work with rods and cubes as the reverse of multiplication. In other words, represent your dividend with, for example 3 ten rods and then find the different rods that will evenly divide this length. A nice lesson in factors and divisibility can come from this. As a student tries out lengths to fit evenly along the 32 unit row, they will be solidifying their knowledge of factors and the concept of division as the partitioning of a number into equal parts.
Re-introducing the number line and the number ray using whole number integers. Work on all four basic operations with whole numbers using the number line. Distance or linear measurement should be stressed. For each of the operations, students have had hands on experience using the rods and cubes, so they should move into this activity nicely. Students can also practice scaling up and down through multiplication and
division on whole numbers. This work should always be done with measurement in mind. For example, if we ask the students to draw a line 8 inches long and we split it into 8 equal pieces, then each piece would need to be 1 inch long. Every 4 pieces would be 4 inches long, etc… If we scale this 8 inch line up by a factor of 3, there will be 24 places on the line and 4 units on the line will be 12 inches long. Linear scaling is a fun and simple lead in to 2 dimensional and 3 dimensional scaling.
Fractions
We can compare, add, subtract, multiply and divide fractions using rods and cubes. Cuisenaire makes rods (discussed in the previous section) with a lengths of 2, 3, 4, etc… up to 10. Experimenting with these operations and having students coming up with the best procedures to solve may be a beneficial experience. I would open up each subsection of the fraction section of this unit with a period (or more if it is productive and beneficial) of rods and cubes. Then I will move into number lines for that particular subsection. In other words, compare fractional values with rods and cubes one day, then compare similar values on a number line the next. Working with arrays when you get to multiplication and division of a fraction by another fraction should be the only departure from this model.
Addition and Subtraction of Fractions with Like Denominators
As we Introduce fractions on the number line will we will need to take time to develop the concept of 1/d (as explained in the background section of this unit). This process can be represented on the number line very effectively. Let's suppose students are to add the like fractions 3/4 and 7/4. To illustrate this using the number line, a starting point would be to begin with the unit fraction Take for example, the unit fraction 1/4, four iterations of which give us the whole. We can continue counting equivalent fractions that represent
wholes by moving along the number line four units at a time until 2 wholes are reached at 8/4, 3 wholes are reached at 12/4, continuing infinitely. To answer the original problem of adding 3/4 and 7/4, students need now only to move from 3/4 seven units to the right to arrive at 10/4.
This is an interesting step for students to experiment with iterations of a unit or general fraction (addition). For example, I might ask the students, "what is 3/4 more than 1/4?" This process of moving a segment of the number line will reinforce length and give a clearer picture of addition. Allow the students to develop number lines that are d parts long, and fill in the iterations until they have 1 whole. Bringing measurement into this process will serve to reinforce concepts of value.
Addition and Subtraction of fractions using the number line follows the placement description above and the background section on equivalence. Given any two fractions with the same denominator, we are just combining the unit parts for addition and comparing the difference in unit parts for subtraction. Students practicing the subdivision of a line, through measurement, in order to add fractions with like denominators will be well prepared for the subdivisions necessary to find common denominators or lowest common multiples.
Addition and Subtraction of Fractions with Unlike Denominators
If given two fractions with different denominators, we will first follow the theorem of equivalence to create like denominators. This process described in the equivalence background section and the background section on adding and subtracting with unlike denominators begins with the basic algebraic principle of multiplying each fraction being added by the denominator of the other fraction. The product created will give you a common denominator to work with. Working with number lines and unlike denominators will be simple when the fractions are related or are obvious factors of the same number. Students will learn to divide a number line into d parts (thirds (in blue) for example) then subdivide each third into fourths (in black), creating a common denominator of twelfths (figure h) from which to work. Now if a student needs to add 1/3 + 1/4 they can find 1/3 as 4/12 on the line and add 1/4 equivalent fraction 3/12 to it, giving you 7/12. Students will know how to subdivide the number line when they use the theorem of equivalence and multiply each fraction by the others denominator. The initial subdivisions can be in thirds or fourths.
It will be a little more complicated when working with denominators that ask students to subdivide a line into 21 evenly spaced parts because the denominators being summed are 3 and 7. At this juncture it is important to ask ourselves, "does the model help in our students understanding, or am I just using it to stay consistent?" If we ground ourselves in the principle of "example sufficiency" 7 during the work with related and simple fractions, we should not need to burden ourselves with a model that may over complicate an idea already embedded.
Finally, students are ready for multiplication of fractions. Using the number line for related fractions, as shown in the multiplication of fractions by fractions background section, and arrays for related and general fractions that do not subdivide so easily, will serve the children well in giving them hands on, contextualized examples of fractions as numbers having value and taking up space.
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