Math 123 Unit 1

Comparing example 1

A child comparing his height to his mother's height.

Sorting example 1

A child goes through her drawer of t-shirts. She puts all the blue shirts in one pile, all the red shirts in another pile, and goes through all her shirts until they have been grouped into piles of the same color.

Comparing example 2

A child playing with a puzzle compares two puzzle pieces. He notices that they are not the same color, and one is a darker green than the other one is.

What two things must be answered in a sorting problem?

A. What is the set of things being sorted? B. What are the categories into which the things are being sorted, or how does the child determine which category to put a particular thing into?

Adding more to one thing to make another thing.

Adding vanilla to cookie dough as a recipe. Adding strawberries to a blender to make a smoothie.

Taking away some from a group of things

Eating a chip off of someone's plate. Giving toys to a friend.

This graph has only two columns. But children can very easily use pictographs with more than two columns. Give a specific example of a graph that relates to the everyday experience of children like this one, but has more than two columns.

Favorite TV show with 3 or more examples, ice cream flavors, birthday month, food, etc.

In order to count properly, children need to understand matching and sequencing clearly. What mistakes might be made by a child trying to count the X's below if s/he does not understand matching and sequencing?

If he doesn't understand he must make a one-to-one match there is no reason in his mind that he has to make sure to touch them all, if he doesn't understand sequencing, he won't realize the X's need to put in an order. Also the X's are scattered.

How do you think a group of children might have actually made the graph themselves?

Give them gridlines to trace with on paper, dog stickers instead of drawing it, and the teacher could have put the stickers there for accuracy.

Sorting example 2

In school, the children sort all the wooden blocks in the classroom. They have a set of boxes and they put all the blocks that are the same size and shape together in the same box.

Can you imagine why a country would not use the metric system?

It would be an inconvenience in changing the system they already have, and would be costly too.

Sequencing example

Lining kids up by height order (tallest to shortest, shortest to tallest, etc.)

How many categories must there be in sorting?

More than one

Matching

Must clearly define two sets of things, and clearly indicate how the child matches one thing from the first set with one thing from the second set (the matching must be one-to-one)

Give three or four other examples of these pairs of "opposite" words that a young child could use to compare things.

Opposite words that a young child could use to compare things would be "taller" and "shorter." You could explain to a child that he or she can use taller and shorter to compare the heights of their friends. The child will be able to say that he or she has certain friends that are taller than the child is, or shorter. Another set of opposite words that a child could use to compare things is "larger" and "smaller" to compare their toys. The child could say that one toy is smaller than another toy, or larger than another toy, in size. Another set of opposite words that a child could use is "greater than" and "less than" when talking about numbers. For example, you could explain to a child that seven is greater than five when teaching them about numbers. A child using these words would help them understand numbers as well.

Taking away from one thing to make another thing

Pouring water back into a pitcher because someone poured too much. Taking off the tomatoes off a sandwich.

Putting two groups of things together to make one group.

Putting a group of toys that are one color with another group that are another color. Putting one child's blocks with another child's blocks to have a bigger set of blocks.

Sequencing example 2

Putting kids in order by their birthday (oldest to youngest).

Putting two things together to make one thing

Putting milk and Nesquik together to make chocolate milk.

Adding more to a group of things.

Putting more clothes into a child's closet to add to what they already have.

Taking a group of things apart to make two groups of things.

Separating toys into different groups. Separating children into different groups to play a game.

What do they use instead?

The Imperial System

A table is set for lunch but there are no napkins at the places. A young child is given a package of napkins and asked to provide every place with a napkin. The child grabs a bunch of napkins from the package and starts to do the task. Explain how the child can eventually be successful even if he or she does not yet know how to count.

The child can be successful even if he or she does not yet know how to count because they can see a pattern of one napkin for each plate. For every plate, there is one napkin, so the child can place one napkin on the place because there is one plate. He or she may not understand and need an adult's help, but once the child is able to recognize this pattern, they would be able to eventually be successful in this task.

A child is given a set of blue counters and a set of red counters. This child also does not yet know how to count, but is asked to tell whether there are more red ones or blue ones. How can s/he do it?

The child can show this visually. While s/he may not be able to count exactly how many there are because the child can't count, it can be shown visually. For example, if there were ten blue counters and seven red ones, the child would be able to show you visually by stacking them, or placing them in rows or columns. Then, visually you would be able to see there were more blue counters because there would be more blue ones in each row over the red ones.

A three-year old child is asked to count the X's below. She points to each X, one at a time, going left to right, saying, "One, two, three, four, five." She is then asked, "How many X's are there in all?" and she seems stumped. What is missing from this child's understanding?

The child doesn't understand how to see things in whole groups, not just individually. Also doesn't understand the phrase "in all" because of this.

Give some examples of what else (besides the length of tables) could be measured with that same standard. The examples should be ones that seem natural to children.

The children could measure the lengths of different classroom objects besides the table such as notebooks, chairs, or a dry erase board.

The children will be doing some baking in class and there are two bags of sugar on a table. The bags are not identical and neither one is tightly filled. The teacher asks them which bag has the most sugar. There is no scale or balance, but there are plenty of kitchen items. What standard of measurement can the children use to answer this question?

The children could use cups as a measurement standard by placing the two bags of sugar into separate bowls and count the cups as they go. Whichever bowl has more cups of sugar, then that will be the one that has the most sugar in the bag, and students will be able to tell the teacher which bag has more sugar in it. The children will also determine volume because it is measuring the amount of sugar in the bag, so this will be finding the volume as well.

There is a table in each of two classrooms in a school. The two tables are different, and a teacher asks some students, "Which table is longer?" Give at least one way the students can give a definite answer to the question. (The tables cannot be moved! They must stay in the two different rooms.)

The students can measure each table using a tape measure. They can compare their lengths and see which one is longer. Or, they can be creative and use other objects such as books or strings to measure.

Now let's imagine there are three tables (A, B, and C) instead of two. One student measures and determines that Table A is longer than Table B, and tells that to the class. Asecond student measures and determines that Table B is longer than Table C. The teacher asks a third student, "Which is longer, Table A or Table C?" Why is the student's job easier than the first two?

The third student's job is easier than the first two because they already know that Table A is longer than Table B, and that Table B is longer than Table C. Since the student has this prior information that was given to them before asking which table was longer, the student's job will be much easier. The first two students did not have as much information.

What is it about the two columns (the "yes" pictures and the "no" pictures) that enables the children to come to this conclusion? In what way does the graph have to be laid out carefully in order that this can be done?

The yes column is higher than the no column and the students wouldn't have to count.

How many countries in the world do not use the metric system?

Three; The United States, Liberia, and Myanmar

What kind of quality is length?

Transitive

In what situations do we say that the opposite of "more" is "less", and in what situations do we say that the opposite of "more" is "fewer"?

We say the opposite of "more" is "less" when we refer to more general entities, such as more and less time, more and less homework, and generally anything else in life that does not regard a specific number. In situations where we say that the opposite of "more" is "fewer" when referring to more specific entities, such as a number of items. For example, we could say that Johnny has fewer apples than Samantha does, and the opposite of that would be that Samantha has more apples than Johnny does.

What quantities besides length and volume might young children measure? Give examples that require different standards of measurement.

Weight, time or temperature with the correct tools in the classroom

Sequencing

What is the set of things being put in order? and How does the child determine what order to put them in? (Size, height, darkness of color, etc. Randomly lining things up would not be a good example.)

What must your comparing examples answer?

a. What two things are being compared? "Comparing" only refers to two things at a time! We use the word this way because children first learn comparison words like longer, taller, darker, softer, higher, more, etc. by comparing only two things. b. What is the characteristic by which they are being compared? (What is the comparison word?)