Equality:
1. When it says in CCSS" "understand the meaning of the equals sign", what are children supposed to understand that it means? (Your explanation should use different vocabulary than "it means that both sides are equal").
It means that both sides are worth the same amount.
2. What is the most common misunderstanding about the equals sign?
That it means the answer comes next.
3. Write a tricky missing number problem that would tell you whether children misunderstood the equal sign or not. Explain what makes the problem tricky.'
Fill in the blank:
2+5=_+4 (this problem is tricky because a blank comes right after an equals sign--checking to see if children think that the equals sign meand the answer comes next.
4. Explain and give examples of what true-false equations are and how they are relevant for discussing the meaning of the equals sign.
They are equations designed to discuss equality and properties of operations. Some useful true equations are:
5=5+0
3+2=2+3
5+1=4+2
A useful false equation is
2+2=4+1
5. Fix these equations so that they mean the same thing, but there are equals signs that don't work:
a. 6 × 7 = 6 × 5 = 30 + 6 = 36 + 6 = 42
6 × 5 = 30
30 + 6 = 36
36 + 6 = 42
6 × 7 = 42
b. 4 × 4 = 16 ÷ 2 = 8
4 × 4 = 16
16 ÷ 2 = 8
Measurement:
6. What is the difference between standard and non standard units? Give examples.
A standard unit is one that everyone has agreed on its size. Examples are inches and cm.
A non-standard unit is anything with a consistent length (or area or weight) that you can compare something to by repeating the same sized unit over and over.
7. What important concepts do children learn from working with non-standard units?
A measurement tells the number of times a unit is repeated. You need to use the same unit over and over (either using the same object multiple times or using objects with identical lengths). Length units need to be put together end to end with no gaps and no overlaps.
8. Why are standard units important? So that people can communicated to each other about a measurement, and everyone is using the same units.
9. Explain what in indirect comparison of lengths is. Give an example.
An indirect comparison is when you take an something and compare its lengths to two other objects, and use those comparisons to compare the two objects. For example, if you know that your computer is longer than a box of cookies, but shorter than a box of cereal, you can say that the box of cookies is shorter than the box of cereal.
10. What would be a good non-standard unit for measuring the length of the whiteboard? Estimate how long the white board is in terms of your non-standard unit.
I'd want to measure the whiteboard using the lid from a math kit. Without measuring, I'd estimate that the whiteboard in our usual classroom is about 8 kit lids long.
11. What would be a good standard measurement tool for finding the circumference of your head? a tape measure
12. What is conservation of number? What is conservation of length? Conservation of volume?
conservation of number is knowing that a set of objects keeps the same number even if the objects are spread out or moved around
Conservation of lenght is knowing that an object like a piece of string keeps the same length, even if it is curled up or laid out striaght
Conservation of volume is knowing that the amount of water doesn't change when you pour it from a fat jar to a skinny jar.
13. What is a common error children use when using a ruler?
Not lining up the end carefully--this can mean that they neglect lining up the ruler with the object, or it can mean that they line up with the wrong point on the ruler (you should line up with the 0-mark on the ruler)
14. Give a benchmark for length in inches, feet, cm and meters.
An inch is about the length of the second joint of my pinky finger, and a credit card is a little more than 2 inches wide.
A foot is a little longer than the length of my shoe, and is a little longer than a standard piece of copy paper.
A cm is a little less than the width of my pinky finger. An inch is about 2 1/2 cm. A unit block from the base 10 blocks set is 1 cm long, and a 10-stick is 10 cm long.
A meter is a little more than a yard, which is about the length from my shoulder to the end of the thumb of my outstretched hand. My waist is about 1 m above the ground.
15. A book is 20 cm across. Is that significantly shorter than the width of your paper (less than 2/3 of the width of your paper) ? About the same as the width of your paper? About the same as the length of your paper? Significantly longer than the length of your paper? Explain how you estimated the answer without measuring (I'm looking for a sensible explanation and a reasonable estimate)
I estimate (from thinking about 2 base 10 sticks end to end) that it's about the width of a sheet of paper. I can check this by thinking that a sheet of paper is about 8 inches wide, and an inch is about 2 1/2 cm, so that would be about 16+4=20 cm (wow! I didn't think it would be that close when I wrote this problem)
16. Write and solve Join or PPW (addition) Separate (take away) and Comparison word problems using length measurements.
Janet knitted 5 inches of a scarf on Monday. On Tuesday, she knitted 4 more inches. How much did she knit on both days together?
I had 60 inches of ribbon. I used 12 inches to investigate area and perimeter. How much ribbon is left?
Adam and Mike pushed their toy cars down a ramp. Adam's car went 12 feet. Mike's car went 9 feet. How much further did Adam's car go that Mike's?
17. Find horizontal and vertical lengths and perimeters on a grid
18. Find areas consisting of whole squares and half-squares on a grid.
19. Estimate areas of irregular shapes on a grid
20. Explain how to find a perimeter of an irregular shape (what tools would you use? What would you do with them?)
I would use something flexible like a piece of string to put around the perimeter, and then I would lay it out straight and measure it with a ruler.
21. Explain how a diagonal length on a grid is different from a horizontal or vertical length.
A diagonal length is longer than either the horizontal or the vertical lengths the line goes across. If you are going over and up by one grid unit each way, the diagonal length will be almost 1 1/2 times as long as the horizontal or vertical length.
22. Explain what the most common errors are when measuring on a grid.
The two most common errors are counting dots instead of spaces to find lengths, and counting diagonal lengths as being the same as horizontal and vertical lengths.
23. Given a fixed area, tell what would be the rectangle with the smallest perimeter
To get the smallest perimeter, you would need the rectangle that would be closest to being square (sides of equal or nearly equal lengths)
24. Given a fixed perimeter, tell what would be the rectangle with the largest area.
Same as the previous one (cool huh?)
25. Given a fixed perimeter, tell how to make a rectangle with a very small area
Make the rectangle very long and skinny: as thin as you can make it.
26. Illustrate the distributive law using rectangle area as a way of thinking about multiplication
27. Explain what makes estimating and comparing weights and volumes tricky.
When the length doubles, the weights and volumes more than double because the object is larger in all 3 dimensions.
28. Write addition, subtraction and multiplication word problems using volume or weight.
My math book weighs 4 lbs, and my science book weighs 3 lbs. How much do my books weigh together?
My bag had 48 ounces of cat food. I fed my cats 5 ounces of cat food. How much cat food is left in the bag?
The jug had 12 cups of milk. The family used 3 cups of milk on cereal. How much milk is left?
A cup holds 7 ounces of juice. How much juice do you need to fill 5 cups?