Big Ideas in Math

1. Big Ideas in Math

2. A teacher in the state of Texas knows that students should be GUIDED but NOT TOLD what to do. A good teacher in the state of Texas knows that children need to construct their own meaning – be active participants in the learning process. A good teacher in the state of Texas knows that the role of the teacher is to allow the students to DISCOVER. A good teacher in the state of Texas allows students to COMMUNICATE with one another by WORKING TOGETHER on cooperative learning activities.

3. A good teacher in the state of Texas does NOT have students watch videos or take timed tests. A good teacher in the state of Texas knows that students should ESTIMATE. A good teacher in the state of Texas allows students to EXPLAIN how they got their answers. When students explain their reasoning they are building METACOGNITION. Metacognition leads to INDEPENDENT LEARNERS.

4. A good teacher in the state of Texas uses EFFECTIVE QUESTIONING STRATEGIES while teaching math. CONVERGENT QUESTIONING – Asks for a specific response (“closed” questions), doesn’t encourage H.O.T.S. Ex.: What is 14 X 25? DIVERGENT QUESTIONING – Many possible answers; provides opportunity for children to think. Example: How did you get your answer?

5. A good teacher in the state of Texas teaches PROBLEM SOLVING STRATEGIES, the GOAL of math instruction. Problem Solving Strategies include: Find a patter Working backwards Draw pictures Make a model Guess and test Try a simple form of the problem A good teacher in the state of Texas uses MANIPULATIVES in math. Manipulatives help students CONSTRUCT MEANING – Manipulatives help students make a connection between concepts and symbols. Technology tools supplement the use of manipulatives.

6. Students in Mr. Lewis class took a survey to determine what their favorite school subjects were. The students showed their preferences by writing their initial in a portion of the Venn Diagram below. Math X C B According to the diagram, what was the total number of students who liked both math and science? Q D G P T W M Z J O S K J L • 1 • 4 • 5 • 8 Science Reading

7. Use the problem to answer the question: X Using each of the numbers 9,8,7,6,5, and 4 exactly once, place one number in each box so that the answer is the largest possible. • This problem would be most appropriate for giving students an opportunity to apply which of the following problem-solving methods? • A. Work backward • B. Guess ad test • C. Draw a diagram • D. Look for a pattern

8. A good teacher in the state of Texas knows some of the vocabulary of math. • ORDERED PAIRS – A pair of numbers that five the location of a point on a graph or a grid. Example: (1,1), (2,3), (3,4)

9. FACT FAMILY – A set of related addition/subtraction or multiplication /division number sentence. Example: 6 + 3 = 9, 3+ 6 = 9 9 - 3 = 6, 9 – 6 = 3 5 x 4 = 20, 4 x 5 = 20 20/4 = 5, 20/5 = 4 INVERSE OPERATION – Two equations with opposite operations that undo each other. Example: 6 + 5 = 1111 – 6 = 5

10. A good teacher in the state of Texas knows how to graph, set up graphs, and what kind of graph to use for a particular set of data. • PICTOGRAPH – A graph that uses pictures to show and COMPARE information. A key tells how many each picture stands for. key

11. LINE GRAPH – A graph that shows change over time. • Circle Graph – a graph that shows parts of a whole and is often indicted in percentages.

12. BAR GRAPH – A graph that compares facts about groups.

13. A good teacher in the state of Texas understand the concept of FUNCTIONS. • Functions are relationships that associate one group of numbers with another. A function shows a PATTERN. • Example:

14. A total of 208 students will be going on a bus trip. If each bus holds 65 students, and there are enough buses for all the students, how many empty seats will be left over? • Two fourth graders decide to use a calculator to solve the problem. They divide 208 by 65 and get an answer of 3.2. The answer confuses them and they ask the teacher what to do next. The teacher’s best initial response would be to: • A. Point out that the answer of 3.2 means that 4 buses will be needed for the trip. • B. Advise the students that their next steps should be to multiply and then subtract. • C. Ask the students to explain why they divided 208 by 65 and what the answer should tell them. • D. Suggest that the students compute the answer by hand and compare that answer to the first one.

15. The teacher wants to help students understand that tables summarize ideas and information. Which of the following strategies would best address this goal? • A. The teacher helps students brainstorm a title for each column, and for the table as a whole, and then writes the final titles on the table. • B. The teacher shows students an example of other simple tables and asks students how the tables are alike. • C. The teacher briefly reviews for students the steps they took to make the table and encourage students to talk about what they have learned. • D. The teacher has students work in small groups to create their own simple tables.

16. A good teacher in the state of Texas knows vocabulary related to GEOMETRY. • Circumference • Diameter • Radius • Angles: Right Angle – 90 • Acute Angle – less than 90 • Obtuse angle – more than 90 • Straight angle - 180 Lines: Line segment Ray Parallel Lines Perpendicular lines.

17. A good teacher in the state of Texas knows that the PYTHAGOREAN THEOREM is: • The square of the hypotenuse of a right triangle is equal to the sum of the square of the other two sides. (the hypotenuse is the side opposite the right triangle.) • A2 + B2 =C2

18. A good teacher in the state of Texas understands the concept of DIMENSION. • FLAT FIGURES have TWO DIMENSIONS and can be measured by LENGTH and WIDTH. • Examples: Triangles, squares, and rectangles. • PLANE – Consists of length and width. • THREE DIMENSIONAL FIGURES (space figures or solid figures) have THREE DIMENSIONS, length, width, and height. • Solid figures can have FACES, EDGES, and VERTICES. The flat surfaces are called faces. • Two faces meet at an edge. Edges meet at vertices. • A CUBE has 6 faces, 12 edges, and 8 vertices.

19. A good teacher in the state of Texas understands the concept of SYMMETRY. • A figure is symmetric if it can be folded along a line so that the 2 parts match exactly. • Example: http://www.adrianbruce.com/Symmetry/whiteboard-activity.htm

20. A good teacher in the state of Texas has an understanding of TRANSFORMATIONS . A transformation refers to the change of position of a figure. FOUR transformations are: reflection, rotation, translation, and dilation.

21. REFLECTION (flip) – The mirror image of a figure that has been flipped over a line of symmetry. • ROTATION – A rotation occurs when the image of a figure is “turned” as if on a wheel. The rotation is made around a point. Example: Turning a door knob. • TRANSLATION – A translation occurs when the image of a figure has been slid to a new position without flipping or turning.

22. DILATION – A dilation changes the size of an object but not its shape. The dilation can make the shape larger or smaller. Example: Honey I shrunk the kids Example: Honey I blew up the kids. CONGURENCE – When two figures have the SAME SIZE AND SAME SHAPE, they are said to be congruent. SIMILAR – When tow figures have the SAME SHAPE but are NOT the SAME SIZE, they are said to be similar.

23. A good teacher in the state of Texas has a basic understanding of probability. PROBABILITY – The chance or likelihood that particular outcome will occur. Example:If a bag has 2 red marbles and 3 purple marbles what is the probability of picking a purple marble? Number of favorable outcome = 3 Total Number of outcomes = 5 Probability is expressed as fraction so the probability of picking a purple marble is 3/5.

24. The students in Ms. Ross’s third grade class are playing a probability game. In a bag are 6 green m&m’s, 4 red m&m’s, 3 yellow m&m’s, and 2 blue m&m’s. A student draws a green m&m from the bag. It is not put back in the bag. What is the most likely outcome for the 2nd draw? • Green • Red • Yellow • Blue

25. The students in Ms. Ross’s third grade class are playing a probability game. In a bag are 6 green m&m’s, 4 red m&m’s, 3 yellow m&m’s, and 2 blue m&m’s. A student draws a green m&m from the bag. It is not put back in the bag. What is the likelihood the 2nd draw will be red? • 1/15 • 1/14 • 4/15 • 2/7

26. Students need to be provided activities to ASSESS their own understanding of math operations. Students should be given opportunities to decide which mathematical operation should be used to solve a particular problem. Mental math skills are required learning experiences.

27. WORK SAMPLE Name: Juan Problem: The moon is 239,256 miles away from the earth. If it takes a spaceship 2 days to go from the earth to the moon how fast did the space ship travel? Use your calculator to solve the problem and explain how you got your answer. Answer 49,845 miles per hour How did you get the answer? First, I figured out how many hours there are in 2 days which is 48 hours. Then I divided the distance to the moon by the time it took the spaceship to get there.

28. Steven, a fourth grade student, used a calculator to solve the previous word problem. When going over Steven’s work with him, the teacher should place the greatest importance on which of the following? • Reminding Steven he should always do each calculation several times whenever he is using a calculator. • Asking Steven to estimate the answer to the problem in order to assess the reasonableness of the answer on the calculator. • Reviewing with Steven the rules for the conversion units within the same system of measurement. • Asking Steven to think of another method to use to solve the problem.